Profiling complex surface structures using scanning interferometry

ABSTRACT

A method including comparing information derivable from a scanning interferometry signal for a first surface location of a test object to information corresponding to multiple models of the test object, wherein the multiple models are parametrized by a series of characteristics for the test object. The derivable information being compared may relate to a shape of the scanning interferometry signal for the first surface location of the test object.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. 119(e) to the followingU.S. Provisional Patent Applications: U.S. Patent Application Ser. No.60/452,615 filed Mar. 6, 2003 and entitled “PROFILING COMPLEX SURFACESTRUCTURES USING HEIGHT SCANNING INTERFEROMETRY,” U.S. PatentApplication Ser. No. 60/452,465 filed Mar. 6, 2003 and entitled“PROFILING COMPLEX SURFACE STRUCTURES USING SIGNALS FROM HEIGHT SCANNINGINTERFEROMETRY,” and U.S. Patent Application Ser. No. 60/539,437 filedJan. 26, 2004 and entitled “SURFACE PROFILING USING AN INTERFERENCEPATTERN MATCHING TEMPLATE,” all of which are incorporated herein byreference.

BACKGROUND

The invention relates to using scanning interferometry to measuresurface topography and/or other characteristics of objects havingcomplex surface structures, such as thin film(s), discrete structures ofdissimilar materials, or discrete structures that are underresolved bythe optical resolution of an interference microscope. Such measurementsare relevant to the characterization of flat panel display components,semiconductor wafer metrology, and in-situ thin film and dissimilarmaterials analysis.

Interferometric techniques are commonly used to measure the profile of asurface of an object. To do so, an interferometer combines a measurementwavefront reflected from the surface of interest with a referencewavefront reflected from a reference surface to produce aninterferogram. Fringes in the interferogram are indicative of spatialvariations between the surface of interest and the reference surface.

A scanning interferometer scans the optical path length difference (OPD)between the reference and measurement legs of the interferometer over arange comparable to, or larger than, the coherence length of theinterfering wavefronts, to produce a scanning interferometry signal foreach camera pixel used to measure the interferogram. A limited coherencelength can be produced, for example, by using a white-light source,which is referred to as scanning white light interferometry (SWLI). Atypical scanning white light interferometry (SWLI) signal is a fewfringes localized near the zero optical path difference (OPD) position.The signal is typically characterized by a sinusoidal carrier modulation(the “fringes”) with bell-shaped fringe-contrast envelope. Theconventional idea underlying SWLI metrology is to make use of thelocalization of the fringes to measure surface profiles.

SWLI processing techniques include two principle trends. The firstapproach is to locate the peak or center of the envelope, assuming thatthis position corresponds to the zero optical path difference (OPD) of atwo-beam interferometer for which one beam reflects from the objectsurface. The second approach is to transform the signal into thefrequency domain and calculate the rate of change of phase withwavelength, assuming that an essentially linear slope is directlyproportional to object position. See, for example, U.S. Pat. No.5,398,113 to Peter de Groot. This latter approach is referred to asFrequency Domain Analysis (FDA).

Unfortunately such assumptions may break down when applied to a testobject having a thin film because of reflections by the top surface andthe underlying film/substrate interface. Recently a method was disclosedin U.S. Pat. No. 6,545,763 to S. W. Kim and G. H. Kim to address suchstructures. The method fit the frequency domain phase profile of a SWLIsignal for the thin film structure to an estimated frequency domainphase profile for various film thicknesses and surface heights. Asimultaneous optimization determined the correct film thickness andsurface height.

SUMMARY

The inventors have realized that there is a wealth of information in ascanning interfometry signal, much of which is ignored in conventionalprocessing. While complex surface structures, such as thin films, maycorrupt conventional processing techniques based on identifying thelocation of the peak in the fringe contrast envelope or calculating aslope for the frequency domain phase profile, new processing techniquesdisclosed herein can extract surface height information and/orinformation about that the complex surface structure.

For example, while not assuming that the surface height information isdirectly related to the peak in the fringe contrast envelope, someembodiments of the invention assume that a change in surface heighttranslates the scanning interferometry signal with respect to areference scan position, but otherwise preserves the shape of thescanning interferometry signal. Thus, the shape of the scanninginterferometry signal is especially useful in characterizing complexsurface structure because it is independent of surface height.Similarly, in the frequency domain, some embodiments assume a change insurface height introduces a linear term in the frequency domain phaseprofile, even though the frequency domain profile itself may not belinear. However, the change in surface height leaves the frequencydomain amplitude profile unchanged. Therefore, the frequency domainamplitude profile is especially useful in characterizing complex surfacestructure.

After the complex surface structure is characterized, surface height canbe efficiently determined. For example, a cross-correlation between thescanning interferometry signal and a model signal having the shapecorresponding to the complex surface structure can produce a peak at ascan coordinate corresponding to the surface height. Similarly, in thefrequency domain, a phase contribution resulting from the complexsurface structure can be subtracted from the frequency domain phaseprofile and the surface height can be extracted using a conventional FDAanalysis.

Examples of complex surface structure include: simple thin films (inwhich case, for example, the variable parameter of interest may be thefilm thickness, the refractive index of the film, the refractive indexof the substrate, or some combination thereof); multilayer thin films;sharp edges and surface features that diffract or otherwise generatecomplex interference effects; unresolved surface roughness; unresolvedsurface features, for example, a sub-wavelength width groove on anotherwise smooth surface; dissimilar materials (for example, the surfacemay comprise a combination of thin film and a solid metal, in which casethe library may include both surface structure types and automaticallyidentify the film or the solid metal by a match to the correspondingfrequency-domain spectra); surface structure that give rise to opticalactivity such as fluorescence; spectroscopic properties of the surface,such as color and wavelength-dependent reflectivity;polarization-dependent properties of the surface; and deflections,vibrations or motions of the surface or deformable surface features thatresult in perturbations of the interference signal.

In some embodiments, the limited coherence length of the light used togenerate the scanning interferometry signal is based on a white lightsource, or more generally, a broadband light source. In otherembodiments, the light source may be monochromatic, and the limitedcoherence length can result from using a high numerical aperture (NA)for directing light to, and/or receiving light from, the test object.The high NA causes light rays to contact the test surface over a rangeof angles, and generates different spatial frequency components in therecorded signal as the OPD is scanned. In yet further embodiments, thelimited coherence can result from a combination of both effects.

The origin of the limited coherence length is also a physical basis forthere being information in the scanning interferometry signal.Specifically, the scanning interferometry signal contains informationabout complex surface structure because it is produced by light rayscontacting the test surface with many different wavelengths and/or atmany different angles.

In the processing techniques described herein, information derivablefrom a scanning interferometry signal for a first surface location of atest object (including the scanning interferometry signal itself) iscompared to information corresponding to multiple models of the testobject, where the multiple models are parametrized by a series ofcharacteristics for the test object. For example, the test object can bemodeled as a thin film and the series of characteristics can be a seriesof values for the thickness of the thin film. While the informationbeing compared might include, for example, information about thefrequency domain phase profile, it might also include information aboutthe shape of the scanning interferometry data and/or information aboutthe frequency domain amplitude profile. Furthermore, to focus thecomparison on the complex surface structure, and not the surface heightat the first surface location, the multiple models can all correspond toa fixed surface height for the test object at the first surfacelocation. The comparison itself can be based on calculating a meritfunction indicative of the similarity between the information from theactual scanning interferometry signal and the information from each ofthe models. For example, the merit function can be indicative of fitbetween the information derivable from the scanning interferometry dataand function parametrized by the series of characteristics.

Furthermore, in some embodiments, the series of characteristicscorresponds to a characteristic of the test object at second locationdifferent from the first location, including for example, diffractivesurface structures that contribute to the interface signal for the firstsurface locations. Thus, while we often refer to the complex surfacestructure as being something other than surface height at the firstsurface location corresponding to the scanning interferometry signal,the complex surface structure may correspond to surface height featuresspaced from the first surface location corresponding to the scanninginterferometry signal.

The methods and techniques described herein can be used for in-processmetrology measurements of semiconductor chips. For example, scanninginterferometry measurements can be used for non-contact surfacetopography measurements semiconductor wafers during chemical mechanicalpolishing (CMP) of a dielectric layer on the wafer. CMP is used tocreate a smooth surface for the dielectric layer, suitable for precisionoptical lithography. Based on the results of the interferometrictopography methods, the process conditions for CMP (e.g., pad pressure,polishing slurry composition, etc.) can be adjusted to keep surfacenon-uniformities within acceptable limits.

We now summarize various aspects and features of the invention.

In general, in one aspect, the invention features a method including:comparing information derivable from a scanning interferometry signalfor a first surface location of a test object to informationcorresponding to multiple models of the test object, wherein themultiple models are parametrized by a series of characteristics for thetest object.

Embodiments of the invention may incude any of the following features.

The method may further include determining an accurate characteristicfor the test object based on the comparison.

The method may further include determining a relative surface height forthe first surface location based on the comparison. Furthermore, thedetermining of the relative surface height may include determining whichmodel corresponds to an accurate one of the characteristic for the testobject based on the comparison, and using the model corresponding to theaccurate characteristic to calculate the relative surface height.

For example, the using of the model corresponding to the accuratecharacteristic may include compensating data from the scanninginterferometry signal to reduce contributions arising from the accuratecharacteristic. The compensating of the data may include removing aphase contribution arising from the accurate characteristic from a phasecomponent of a transform of the scanning interferometry signal for thetest object, and the using of the model corresponding to the accuratecharacteristic may further include calculating the relative surfaceheight from the phase component of the transform after the phasecontribution arising from the accurate characteristic has been removed.

In another example, using the model corresponding to the accuratecharacteristic to calculate the relative surface height may includedetermining a position of a peak in a correlation function used tocompare the information for the test object to the information for themodel corresponding to the accurate characteristic.

The method may further include comparing information derivable from thescanning interferometry signal for additional surface locations to theinformation corresponding to the multiple models. Also, the method mayfurther include determining a surface height profile for the test objectbased on the comparisons.

The comparing may include calculating one or more merit functionsindicative of a similarity between the information derivable from thescanning interferometry signal and the information corresponding to eachof the models.

The comparing may include fitting the information derivable from thescanning interferometry signal to an expression for the informationcorresponding to the models.

The information corresponding to the multiple models may includeinformation about at least one amplitude component of a transform (e.g.,a Fourier transform) of a scanning interferometry signal correspondingto each of the models of the test object. Likewise, the informationderivable from the scanning interferometry signal includes informationabout at least one amplitude component of a transform of the scanninginterferometry signal for the test object.

The comparing may include comparing a relative strength of the at leastone amplitude component for the test object to the relative strength ofthe at least one amplitude component for each of the models.

The information corresponding to the multiple models may be a functionof a coordinate for the transform. For example, the informationcorresponding to the multiple models may include an amplitude profile ofthe transform for each of the models. Furthemore, the comparing mayinclude comparing an amplitude profile of a transform of the scanninginterferometry signal for the test object to each of the amplitudeprofiles for the models.

The comparing may also include comparing information in a phase profileof the transform of the scanning interferometry signal for the testobject to information in a phase profilde of the transform for each ofthe models. For example, the information in the phase profiles mayinclude information about nonlinearity of the phase profile with respectto the transform coordinate and/or information about a phase gap value.

The information derivable from the scanning interferometry signal andwhich is being compared may be a number. Alternatively, the informationderivable from the scanning interferometry signal and which is beingcompared may be a function. For example, it may be a function of scanposition or a function of spatial frequency.

The information for the test object may be derived from a transform(e.g., a Fourier transform) of the scanning interferometry signal forthe test object into a spatial frequency domain. The information for thetest object may include information about an amplitude profile of thetransform and/or a phase profile of the transform.

The information for the test object may relate to a shape of thescanning interferometry signal for the test object at the firstlocation. For example, the information for the test object may relate toa fringe contrast magnitude in the shape of the scanning interferometrysignal. It may also relate to a relative spacings between zero-crossingsin the shape of the scanning interferometry signal. It may also beexpressed as a function of scan position, wherein the function isderived from the shape of the scanning interferometry signal.

The comparing may include calculating a correlation function (e.g., acomplex correlation function) between the information for the testobject and the information for each of the models. The comparing mayfurther include determining one or more peak values in each of thecorrelation functions. The method may then further include determiningan accurate characteristic for the test object based on theparameterization of the model corresponding to the largest peak value.Alternately, or in addition, the method may further include determininga relative surface height for the test object at the first surfacelocation based on a coordinate for at least one of the peak values inthe correlation functions.

The multiple models may correspond to a fixed surface height for thetest object at the first location.

The series of characteristics may include a series of values for atleast one physical parameter of the test object. For example, the testobject may include a thin film layer having a thickness, and thephysical parameter may be the thickness of the thin film at the firstlocation.

The series of characteristics may include a series of characteristics ofthe test object at a second surface location different from the firstsurface location. For example, the test object may include structure atthe second surface location that diffracts light to contribute to thescanning interferometry signal for the first surface location. In oneexample, the series of characteristics at the second surface locationmay include permutations of a magnitude for a step height at the secondlocation and a position for the second location. In another example, theseries of characteristics at the second surface location may includepermutations of a modulation depth for a grating and an offset positionof the grating, wherein the grating extends over the second location.

The series of characteristics may be a series of surface materials forthe test object.

The series of characteristics may be a series of surface layerconfigurations for the test object.

The scanning interferometry signal may be produced by a scanninginterferometry system, and the comparing may include accounting forsystematic contributions to the scanning interferometry signal arisingfrom the scanning interferometry system. For example, the systematiccontributions may include information about a dispersion in a phasechange on reflection from components of the scanning interferometrysystem. Furthermore, the method may also include comparing informationderivable from the scanning interferometry signal for additional surfacelocations to the information corresponding to the multiple models, inwhich case, the systematic contributions may be resolved for multipleones of the surface locations. The method may further includecalibrating the systematic contributions of the scanning interferometrysystem using another test object having known properties.

The scanning interferometry signal may produced by imaging test lightemerging from the test object to interfere with reference light on adetector, and varying an optical path length difference from a commonsource to the detector between interfering portions of the test andreference light, wherein the test and reference light are derived fromthe common source (e.g., a spatially extended source), and wherein thescanning interferometry signal corresponds to an interference intensitymeasured by the detector as the optical path length difference isvaried.

The test and reference light may have a spectral bandwidth greater thanabout 5% of a central frequency for the test and reference light.

The common source may have a spectral coherence length, and the opticalpath length difference is varied over a range larger than the spectralcoherence length to produce the scanning interferometry signal.

Optics used to direct test light onto the test object and image it tothe detector may define a numerical aperture for the test light greaterthan about 0.8.

The method may further include producing the scanning interferometrysignal.

In another aspect, the invention features an apparatus including: acomputer readable medium having a program that causes a processor in acomputer to compare information derivable from a scanning interferometrysignal for a first surface location of a test object to informationcorresponding to multiple models for the test object, wherein themultiple models are parametrized by a series of characteristics for thetest object.

The apparatus may include any of the features described above inconnection with the method.

In another aspect, the invention features an apparatus including: ascanning interferometry system configured to produce a scanninginterferometry signal; and an electronic processor coupled to thescanning interferometry system to receive the scanning interferometrysignal and programmed to compare information derivable from a scanninginterferometry signal for a first surface location of a test object toinformation corresponding to multiple models of the test object, whereinthe multiple models are parametrized by a series of characteristics forthe test object.

The apparatus may include any of the features described above inconnection with the method.

In general, in another aspect, the invention features a methodincluding: chemically mechanically polishing a test object; collectingscanning interferometry data for a surface topography of the testobject; and adjusting process conditions for the chemically mechanicallypolishing of the test object based on information derived from thescanning interferometry data. For example, the process conditions may bepad pressure and/or polishing slurry composition. In preferredembodiments, adjusting the process conditions based on the informationderived from the scanning interferometry data may include comparinginformation derivable from the scanning interferometry signal for atleast a first surface location of a test object to informationcorresponding to multiple models of the test object, wherein themultiple models are parametrized by a series of characteristics for thetest object. Analysis of the scanning interferometry signal may furtherinclude any of the features described above with the first-mentionedmethod.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. In case of conflict withpublications, patent applications, patents, and other referencesmentioned incorporated herein by reference, the present specification,including definitions, will control.

Other features, objects, and advantages of the invention will beapparent from the following detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of an interferometry method.

FIG. 2 is a flow chart showing a variation of the interferometry methodof FIG. 1.

FIG. 3 is a schematic drawing of a Linnik-type scanning interferometer.

FIG. 4 is a schematic drawing of a Mirau-type scanning interferometer.

FIG. 5 is a diagram showing illumination of the test sample through anobjective lens.

FIG. 6 shows theoretical Fourier amplitude spectra for scanninginterferometry data in two limits.

FIG. 7 shows two surface types, with and without a thin film

FIG. 8 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 0.

FIG. 8 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 0.

FIG. 9 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 50 nm.

FIG. 10 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 100 nm.

FIG. 11 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 300 nm.

FIG. 12 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 600 nm.

FIG. 13 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 1200 nm.

FIG. 14 shows the surface and substrate profiles determined for asimulation of a SiO₂ on Si thin film in which the film thickness variesuniformly from 0 to 1500 nm in 10-nm increments per pixel, with the topsurface always at zero.

FIG. 15 shows the surface and substrate profiles determined for asimulation identical to that in FIG. 14 except that random noise hasbeen added (2-bits rms out of an average 128 intensity bits).

FIG. 16 shows surface height profiles determined using conventional FDAanalysis (FIG. 16 a) and a library search method described herein (FIG.16 b) for a 2400 line per mm grating having an actual peak-to-valleymodulation depth of 120 nm.

FIG. 17 shows distortions caused by an under-resolved step height in ascanning interference signals for pixels corresponding to varioussurface locations near the step height.

FIG. 18 shows nonlinear distortions in the frequency domain phasespectra for pixels corresponding to surface locations to the left (FIG.18 a) and the right (FIG. 18 b) of the under-resolved step height inFIG. 17.

FIG. 19 shows surface height profiles determined using conventional FDAanalysis (FIG. 19 a) and a library search method described herein (FIG.1 b) for an under-resolved step height.

FIG. 20 shows an actual scanning interferometry signal of a base Sisubstrate without a thin film.

FIGS. 21 and 22 show interference template patterns for a bare Sisubstrate and a thin film structure with 1 micron of SiO2 on Si,respectively.

FIGS. 23 and 24 show the merit function as a function of scan positionsfor template functions in FIGS. 21 and 22, respectively.

Like reference numerals in different drawings refer to common elements.

DETAILED DESCRIPTION

FIG. 1 shows a flow chart that generally describes one embodiment of theinvention in which the analysis of the scanning interferometry data isperformed in the spatial frequency domain.

Referring to FIG. 1, to measure data from a test object surface aninterferometer is used to mechanically or electro-optically scan theoptical path difference (OPD) between a reference and measurement path,the measurement path being directed to an object surface. The OPD at thebeginning of the scan is a function of the local height of the objectsurface. A computer records an interference intensity signal during theOPD scan for each of multiple camera pixels corresponding to differentsurface locations of the object surface. Next, after storinginterference intensity signal as a function of OPD scan position foreach of the different surface locations, the computer performs atransform (e.g., a Fourier Transform) to generate a frequency-domainspectrum of the signal. The spectrum contains both magnitude and phaseinformation as a function of the spatial frequency of the signal in thescanning dimension. For example, a suitable frequency domain analysis(FDA) for generating such a spectrum is disclosed in commonly owned U.S.Pat. No. 5,398,113 by Peter de Groot and entitled “Method and Apparatusfor Surface Topography Measurements by Spatial-Frequency Analysis ofInterferograms,” the contents of which are incorporated herein byreference.

In a separate step, the computer generates a library of theoreticalpredictions for frequency-domain spectra for a variety of surfaceparameters and a model for the interferometer. These spectra may forexample cover a range of possible thin film thicknesses, surfacematerials, and surface textures. In preferred embodiments, the computergenerates library spectra for a constant surface height, e.g.height=zero. Thus, in such embodiments, the library contains noinformation regarding surface topography, only information relative tothe type of surface structure and the interaction of this surfacestructure, the optical system, the illumination, and detection systemwhen generating distinctive features of the frequency-domain spectra. Asan alternative, the prediction library may be generated empirically,using sample artifacts. As another alternative, the library may useinformation from prior supplemental measurements of the object surfaceprovided by other instruments, for example an ellipsometer, and anyother input from a user regarding known properties of the objectsurface, so as to reduce the number of unknown surface parameters. Anyof these techniques for library creation, theoretical modeling,empirical data, or theory augmented by supplemental measurements, may beexpanded by interpolation to generate intermediate values, either aspart of the library creation or in real time during a library search.

In a next step, the experimental data is compared to the predictionlibrary by means of a library search that provides surface structureparameters. In the example case of a film of unknown thickness, thelibrary for a single surface type, e.g. SiO₂ on Si, would range overmany possible film thicknesses with the top surface height always equalto zero. Another example case would be surface roughness, for which theadjustable parameter may be roughness depth and/or spatial frequency.The library search leads to a match to those characteristics of the FDAspectra that are independent of surface height, for example, the averagevalue of the magnitude spectrum, which is related to the overallreflectivity of the surface, or the variation in magnitude as a functionof spatial frequency, which in a monochromatic high-NA system relates tothe scattering angle of the reflected light.

The analysis may also include a system characterization, which includes,e.g. measuring one or more reference artifacts having a known surfacestructure and surface topography, so as to determine parameters such assystem wavefront error, dispersion, and efficiency that may not beincluded in the theoretical model.

Furthermore, the analysis may include an overall calibration, whichincludes e.g., measuring one or more reference artifacts to determinethe correlation between measured surface parameters, such as filmthickness as determined by the library search, and the values for theseparameters as determined independently, e.g. by ellipsometric analysis.

Based on the comparison of the experimental data to the predictionlibrary, the computer identifies the surface model corresponding to thebest match. It may then displays or transmits surface parameter resultsnumerically or graphically to the user or to a host system for furtheranalysis or for data storage. Using the surface parameter results, thecomputer may then determine surface height information in addition tocharacteristics identified by the library search. In some embodiments,the computer generates a compensated phase spectrum, for example bysubtracting the corresponding theoretical phase spectrum directly fromthe experimental phase spectrum. The computer then determines the localsurface height for one or more surface points by analysis of thecompensated phase as a function of spatial frequency, for example byanalysis of the coefficients generated by a linear fit. Thereafter, thecomputer generates a complete three-dimensional image constructed fromthe height data and corresponding image plane coordinates, together withgraphical or numerical display of the surface characteristics asdetermined by the library search.

In some cases, the library search and data collection can be performediteratively to further improve the results. Specifically, the librarysearch can be refined on a pixel-by-pixel or regional basis, by thecreation of refined libraries relevant to the local surface type. Forexample, if it is found that the surface has a thin film ofapproximately 1 micron during a preliminary library search, then thecomputer may generate a fine-grain library of example values close to 1micron to further refine the search.

In further embodiments, the user may only be interested in the surfacecharacteristics modeled by the prediction library, but not surfaceheight, in which case the steps for determining surface height are notperformed. Conversely, the user may only be interested in surfaceheight, but not the surface characteristics modeled in the predictionlibrary, in which case the computer uses the comparison between theexperimental data and the prediction library to compensate theexperimental data for the contributions of the surface characteristics,so that the surface height is more accurately determined, but need notexplicitly determine the surface characteristics or display them.

The analysis may be applied to a variety of surface analysis problems,including: simple thin films (in which case, for example, the variableparameter of interest may be the film thickness, the refractive index ofthe film, the refractive index of the substrate, or some combinationthereof); multilayer thin films; sharp edges and surface features thatdiffract or otherwise generate complex interference effects; unresolvedsurface roughness; unresolved surface features, for example, asub-wavelength width groove on an otherwise smooth surface; dissimilarmaterials (for example, the surface may comprise a combination of thinfilm and a solid metal, in which case the library may include bothsurface structure types and automatically identify the film or the solidmetal by a match to the corresponding frequency-domain spectra); opticalactivity such as fluorescence; spectroscopic properties of the surface,such as color and wavelength-dependent reflectivity;polarization-dependent properties of the surface; deflections,vibrations or motions of the surface or deformable surface features thatresult in perturbations of the interference signal; and data distortionsrelated to the data acquisition procedure, e.g. a data acquisitionwindow that does not fully encompass the interference intensity data.

The interferometer may include any of the following features: aspectrally narrow-band light source with a high numerical aperture (NA)objective; a spectrally broad band light source; a combination of a highNA objective and a spectrally broadband source; an interferometricmicroscope objectives, including oil/water immersion and solid immersiontypes, in e.g. Michelson, Mirau or Linnik geometries; a sequence ofmeasurements at multiple wavelengths; unpolarized light; and polarizedlight, including linear, circular, or structured. For example,structured polarized light may involve, for example, a polarizationmask, generating different polarizations for different segments of theillumination or imaging pupils, so as to reveal polarization-dependentoptical effects attributable to surface characteristics. Theinterferometer may also include the overall system calibration,described above.

In comparing the theoretical and experimental data, the library searchmay be based on any of the following: a product of, or a differencebetween, magnitude and/or phase data in the frequency spectrum,including, e.g., the product of, or difference between, the averagemagnitude and the average phase, the average magnitude itself, and theaverage phase itself; the slope, width and/or height of the magnitudespectrum; interference contrast; data in the frequency spectrum at DC orzero spatial frequency; nonlinearity or shape of the magnitude spectrum;the zero-frequency intercept of the phase; nonlinearity or shape of thephase spectrum; and any combination of these criteria. Note that as usedherein magnitude and amplitude are used interchangeably.

FIG. 2 shows a flow chart that generally describes another embodimentfor the analysis of scanning interferometry data. The analysis issimilar to that described for FIG. 1 except that comparison between theexperimental data and the prediction library is based on information inscan coordinate domain. The experimentl signal may be characterized by aquasi-periodic carrier oscillation modulated in amplitude by an envelopefunction with respect to the scan coordinate. In comparing thetheoretical and experimental data, the library search may be based onany of the following: average signal strength; the shape of the signalenvelope, including e.g. deviation from some ideal or reference shapesuch as a gaussian; the phase of the carrier signal with respect to theenvelope function; the relative spacing of zero crossings and/or signalmaxima and minima; values for maxima and minima and their ordering; peakvalue of the correlation between the library and measured signals, afteradjusting for optimal relative scan position; and any combination ofthese criteria.

In what follows we provide a detailed mathematical description of theanalyses and provide examples. First, we describe exemplary scanninginterferometers. Second, we determine a mathematical model for scanninginterferometry data. Third, we describe optical properties of surfacesand how to use such information to generate accurate models of scanninginterferometry data for different surface characteristics. Fourth, wedescribe how experimental interferometry data can be compared to theprediction library to provide information about the test object.Initially, we will describe thin film applications, and later we willdescribe applications to other complex surface structures, specifically,optically under-resolved step heights and grating patterns. Also, wewill initially focus on analyses in the spatial frequency domain, andlater we will describe analyses in the scan coordinate domain.

FIG. 3 shows a scanning interferometer of the Linnik type. Illuminationlight 102 from a source (not shown) is partially transmitted by a beamsplitter 104 to define reference light 106 and partially reflected bybeam splitter 104 to define measurement light 108. The measurement lightis focused by a measurement objective 110 onto a test sample 112 (e.g.,a sample comprising a thin single- or multi-layer film of one or moredissimilar materials). Similarly, the reference light is focused by areference objective 114 onto a reference mirror 116. Preferably, themeasurement and reference objectives have common optical properties(e.g., matched numerical apertures). Measurement light reflected (orscattered or diffracted) from the test sample 112 propagates backthrough measurement objective 110, is transmitted by beam splitter 104,and imaged by imaging lens 118 onto a detector 120. Similarly, referencelight reflected from reference mirror 116 propagates back throughreference objective 114, is reflected by beam splitter 104, and imagedby imaging lens 118 onto a detector 120, where it interferes with themeasurement light.

For simplicity, FIG. 3 shows the measurement and reference lightfocusing onto particular points on the test sample and reference mirror,respectively, and subsequently interfering on a corresponding point onthe detector. Such light corresponds to those portions of theillumination light that propagate perpendicular to the pupil planes forthe measurement and reference legs of the interferometer. Other portionsof the illumination light ultimately illuminate other points on the testsample and reference mirror, which are then imaged onto correspondingpoints on the detector. In FIG. 3, this is illustrated by the dashedlines 122, which correspond to the chief rays emerging from differentpoints on the test sample that are imaged to corresponding points on thedetector. The chief rays intersect in the center of the pupil plane 124of the measurement leg, which is the back focal plane of measurementobjective 110. Light emerging from the test sample at an angle differentfrom that of the chief rays intersect at a different location of pupilplane 124.

In preferred embodiments, detector 120 is a multiple element (i.e.,multi-pixel) camera to independently measure the interference betweenthe measurement and reference light corresponding to different points onthe test sample and reference mirror (i.e., to provide spatialresolution for the interference pattern).

A scanning stage 126 coupled to test sample 112 scans the position ofthe test sample relative to measurement objective 110, as denoted by thescan coordinate ζ in FIG. 3. For example, the scanning stage can bebased on a piezoelectric transducer (PZT). Detector 120 measures theintensity of the optical interference at one or more pixels of thedetector as the relative position of the test sample is being scannedand sends that information to a computer 128 for analysis.

Because the scanning occurs in a region where the measurement light isbeing focused onto the test sample, the scan varies the optical pathlength of the measurement light from the source to the detectordifferently depending on the angle of the measurement light incident on,and emerging from, the test sample. As a result, the optical pathdifference (OPD) from the source to the detector between interferingportions of the measurement and reference light scale differently withthe scan coordinate ζ depending on the angle of the measurement lightincident on, and emerging from, the test sample. In other embodiments ofthe invention, the same result can be achieved by scanning the positionof reference mirror 116 relative to reference objective 114 (instead ofscanning test sample 112 realtive to measurement objective 110).

This difference in how OPD varies with the scan coordinate ζ introducesa limited coherence length in the interference signal measured at eachpixel of the detector. For example, the interference signal (as afunction of scan coordinate) is typically modulated by an envelopehaving a spatial coherence length on the order of λ/2 (NA)², where λ isthe nominal wavelength of the illumination light and NA is the numericalaperture of the measurement and reference objectives. As describedfurther below, the modulation of the interference signal providesangle-dependent information about the reflectivity of the test sample.To increase the limited spatial coherence, the objectives in thescanning interferometer preferably define a large numerical aperture,e.g., greater than about 0.7 (or more preferably, greater than about0.8, or greater than about 0.9). The interference signal can also bemodulated by a limited temporal coherence length associated with thespectral bandwidth of the illumination source. Depending on theconfiguration of the interferometer, one or the other of these limitedcoherence length effects may dominate, or they may both contributesubstantially to the overall coherence length.

Another example of a scanning interferometer is the Mirau-typeinterferometer shown in FIG. 4.

Referring to FIG. 4, a source module 205 provides illumination light 206to a beam splitter 208, which directs it to a Mirau interferometricobjective assembly 210. Assembly 210 includes an objective lens 211, areference flat 212 having a reflective coating on a small centralportion thereof defining a reference mirror 215, and a beam splitter213. During operation, objective lens 211 focuses the illumination lighttowards a test sample 220 through reference flat 212. Beam splitter 213reflects a first portion of the focusing light to reference mirror 215to define reference light 222 and transmits a second portion of thefocusing light to test sample 220 to define measurement light 224. Then,beam splitter 213 recombines the measurement light reflected (orscattered) from test sample 220 with reference light reflected fromreference mirror 215, and objective 211 and imaging lens 230 image thecombined light to interfere on detector (e.g., a multi-pixel camera)240. As in the system of FIG. 3, the measurement signal(s) from thedetector is sent to a computer (not shown).

The scanning in the embodiment of FIG. 4 involves a piezoelectrictransducer (PZT) 260 coupled to Mirau interferometric objective assembly210, which is configured to scan assembly 210 as a whole relative totest sample 220 along the optical axis of objective 211 to provide thescanning interferometry data I(ζ, h) at each pixel of the camera.Alternatively, the PZT may be coupled to the test sample rather thanassembly 210 to provide the relative motion there between, as indicatedby PZT actuator 270. In yet further embodiments, the scanning may beprovided by moving one or both of reference mirror 215 and beam splitter213 relative to objective 211 along the optical axis of objective 211.

Source module 205 includes a spatially extended source 201, a telescopeformed by lenses 202 and 203, and a stop 204 positioned in the frontfocal plane of lens 202 (which coincides with the back focal plane oflens 203). This arrangement images the spatially extended to source ontothe pupil plane 245 of Mirau interferometric objective assembly 210,which is an example of Koehler imaging. The size of stop controls thesize of the illumination field on test sample 220. In other embodiments,the source module may include an arrangement in which a spatiallyextended source is imaged directly onto the test sample, which is knownas critical imaging. Either type of source module may be used with theLinnik-type scanning interferometry system of FIG. 1.

In further embodiments of the invention, the scanning interferometrysystem may used to determine angle-dependent scattering or diffractioninformation about a test sample, i.e., for scatterometry. For example,the scanning interferometry system may be used to illuminate a testsample with test incident over only a very narrow range of incidentangles (e.g., substantially normal incidence or otherwise collimated),which may then be scattered or diffracted by the test sample. The lightemerging from the sample is imaged to a camera to interfere withreference light as described above. The spatial frequency of eachcomponent in the scanning interferometry signal will depend vary withangle of the test light emerging from the test sample. Thus, a verticalscan (i.e., a scan along the optical axis of an objective) followed byFourier analysis allows for a measurement of diffracted and/or scatteredlight as a function of emerging angle, without directly accessing orimaging the back focal plane of the objective. To provide thesubstantially normal incidence illumination, for example, the sourcemodule can be configured to image a point source onto the pupil plane orto otherwise decrease the degree to which the illumination light fillsthe numerical aperature of the measurement objective. The scatterometrytechnique may be useful for resolving discrete structures in the samplesurface, such as grating lines, edges, or general surface roughness,which may diffract and/or scatter light to higher angles.

In much of the analysis herein, it is assumed that the polarizationstate of the light in the pupil plane is random, i.e., comprised ofapproximately equal amounts of both s polarizations (orthogonal to theplane of incidence) and p (orthogonal to the plane of incidence)polarizations. Alternative polarizations are possible, including pure spolarization, such as may be realized by means of a radial polarizerplaced in the pupil plane (e.g., in the back-focal plane of themeasurement object in the case of a Linnik interferometer and in theback focal plane of the common objective in the Mirau interferometer).Other possible polarizations include radial p polarization, circularpolarization, and modulated (e.g. two states, one following the other)polarization for ellipsometric measurements. In other words, opticalproperties of the test sample can be resolved not only with respect totheir angle- or wavelength-dependence, but also with respect to theirpolarization dependence or with respect to a selected polarization. Suchinformation may also be used to improve the accuracy of thin filmstructure characterization.

To provide such ellipsometry measurements, the scanning interferometrysystem may include a fixed or variable polarizer in the pupil plane.Referring again to FIG. 4, the Mirau-type interferometry system, forexample, includes polarization optics 280 in the pupil plane to select adesired polarization for the ligh incident on, and emerging from thetest sample. Furthermore, the polarization optics may be reconfigurableto vary the selected polarization. The polarization optics may includeone or more elements including polarizers, waveplates, apodizationapertures, and/or modulation elements for selecting a givenpolarization. Furthermore, the polarization optics may be fixed,structured or reconfigurable, for the purpose of generating data similarto that of an ellipsometer. For example, a first measurement with aradially-polarized pupil for s polarization, followed by aradially-polarized pupil for p polarization. In another example, one mayuse an apodized pupil plane with linearly polarized light, e.g., a slitor wedge, which can be rotated in the pupil plane so as to direct anydesired linear polarization state to the object, or a reconfigurablescreen such as a liquid crystal display.

Moreover, the polarization optics may provide a variable polarizationacross the pupil plane (e.g., by including multiple polarizers or aspatial modulator). Thus, one can “tag” the polarization state accordingto spatial frequency, for example, by providing a different polarizationfor high angles of incidence than shallow angles.

In yet further embodiments, the selectable polarization may be combinedwith a phase shift as a function of polarization. For example, thepolarization optics may include a linear polarizer is positioned in thepupil plane and followed by two waveplates (e.g., eighth-wave plates) inopposing quadrants of the pupil plane. The linear polarization resultsin a full range of polarization angles with respect to the incidentplanes of the objective. If the waveplates are aligned so that, forexample, the predominately s-polarized light has a fixed phase shift,then both radial s polarized and p polarized light are presentsimultaneously, but shifted in phase with respect to each other, e.g.,by pi, so that the interferometer is effectively detecting thedifference between these two polarization states as the fundamentalsignal.

In further embodiments, polarization optics may be positioned elsewherein the apparatus. For example, linear polarization can be achievedanywhere in the system.

We now describe a physical model for the scanning interferometry signal.

The object surface has height features h which we wish to profile overan area indexed by lateral coordinates x,y. The stage provides a smooth,continuous scan ζ either of the interference objective or, as shown, ofthe object itself. During the scan, a computer records intensity dataI_(ζ,h) for each image point or camera pixel in successive cameraframes. Note that the key dependencies of the intensity I_(ζ,h) on thescan position and surface height are indicated by subscripts—a notationthat we shall employ throughout.

A proper physical model of the optics can be very elaborate, taking intoaccount the partial coherence of the light source, polarization mixingin the interferometer, the imaging properties of high-NA objectives, andthe interaction of electric field vectors at high angles of incidenceand in the presence of discontinuous surface features. For convenience,we simplify the model by assuming random polarization and diffuse,low-coherence extended sources. Modeling the interference signalsimplifies to adding up the contributions of all of the ray bundlespassing through the pupil plane of the objective and reflecting from theobject surface at an incident angle ψ, as shown in FIG. 5.

The interference contribution for a single ray bundle through theoptical system is proportional tog _(β,k,ζ,h) =R _(β,k) +Z _(β,k) +2{square root}{square root over (R_(β,k) Z _(β,k) )} cos [2βkn ₀(h−ζ)+(υ_(β,k)−ω_(β,k))]  (1)Where Z_(β,k) is the effective object intensity reflectivity, includinge.g. the effects of the beamsplitter, and R_(β,k) is the effectivereference reflectivity, including both the beamsplitter and thereference mirror. The index of the ambient medium is n₀, the directionalcosine for an incident angle ψ isβ=cos(ψ)  (2)and the wavenumber for the source illumination isk=(2π/λ)  (3)The sign convention for the phase causes an increase in surface heightto correspond to a positive change in phase. The phase term has acontribution ω_(β,k) for the object path in the interferometer,including thin film effects from the object surface, and a contributionυ_(β,k) for the reference path, including the reference mirror and otheroptics in the objective.

The total interference signal integrated over the pupil plane isproportional to $\begin{matrix}{I_{\zeta,h} = {\int_{0}^{\infty}{\int_{0}^{1}{g_{\beta,k,\zeta,h}\quad U_{\beta}V_{k}\beta\quad{\mathbb{d}\beta}\quad{\mathbb{d}k}}}}} & (4)\end{matrix}$where U_(β) is the pupil plane light distribution and V_(k) the opticalspectrum distribution. The weighting factor β in Eq. (4) follows from acos(ψ) term attributable to the projection angle and a sin(ψ) term forthe diameter of the annulus of width dψ in the pupil plane:cos(ψ)sin(ψ)dψ=−βdβ  (5)Here we assume that the objective obeys the Abbé sine condition as shownin FIG. 5. This relatively simple weighting is possible for randomlypolarized, spatially incoherent illumination, for which all ray bundlesare independent from each other. Finally, the integration limits overall incident angles implies 0≦β≦1 and the spectrum integration over allwavenumbers 0≦k≦∞.

In a frequency domain analysis (FDA), we first calculate the FourierTransform of the interference intensity signal I_(ζ,h). For the literal(non-numerical) analysis we shall use the un-normalized Fourier integral$\begin{matrix}{q_{Κ,h} = {\int_{- \infty}^{\infty}{I_{\zeta,h}\quad{\exp\left( {{\mathbb{i}}\quad Κ\quad\zeta} \right)}\quad{\mathbb{d}\zeta}}}} & (6)\end{matrix}$where K is the spatial frequency, e.g. in cycles per μm. Thefrequency-domain value q_(K,h) carries units of inverse wavenumber, e.g.μm. From this there follows a power spectrumQ_(K,h) =|q _(K,h)|²  (7)and a phase spectrumφ″_(K,h) =arg(q _(K,h)).  (8)The double prime for φ″_(K,h) means that there is a two-fold uncertaintyin the fringe order, both from pixel to pixel and overall with respectto the starting point in the scan. Conventional FDA then proceedsdirectly to a determination of surface topography by a linear fit to thephase spectrum φ″_(K,h) weighted by the power spectrum Q_(K,h). The fitprovides for each pixel a slopeσ_(h) ≈dφ″/dK  (9)and an interceptA″≈φ″_(K=0,h).  (10)Note that the intercept or “phase gap” A″ is independent of height h,but carries the double prime inherited from the fringe order uncertaintyin the phase data. The slope σ is free of this uncertainty. From theintercept A″ and the slope σ_(h), we can define for a specific mean ornominal spatial frequency K0 a “coherence profile”Θ_(h)=σ_(h) K0  (11)and a “phase profile”θ″_(h)=Θ_(h) +A″.  (12)For the simple, idealized case of a perfectly uniform, homogeneousdielectric surface free of thin films and dissimilar material effects,and an optical system perfectly balanced for dispersion, the phase andcoherence profiles are linearly proportional to surface height:h _(Θ)=Θ_(h) /K0  (13)h″ _(θ)=θ″_(h) /K0  (14)Of the two height calculations, the height value h″_(θ) based on phaseis the more accurate, but it has the uncertainty in the fringe ordercharacteristic of monochromatic interferometry. For high resolution, wecan use the unambiguous but less precise height value h_(Θ) based oncoherence to remove this uncertainty and yield a final value h_(θ).

Conventional FDA assumes that even for less idealized situations, theinterference phase φ″_(K,h) is still nearly a linear function of spatialfrequency. For the present embodiment, however, we determine keyparameters of the surface structure such as film thickness by comparingexperimental data to a theoretical prediction that may include highlynonlinear phase spectra and associated modulations of the powerspectrum.

To this end, we combine the definition of the Fourier Transform Eq. (6)with the interference signal Eq. (4) into the following formula for thepredicted FDA spectrum: $\begin{matrix}{q_{Κ,h} = {\int_{- \infty}^{\infty}{\int_{0}^{\infty}{\int_{0}^{1}{g_{\beta,k,\zeta,h}\quad{\exp\left( {{\mathbb{i}}\quad Κ\quad\zeta} \right)}\quad U_{\beta}V_{k}\beta\quad{\mathbb{d}\beta}\quad{\mathbb{d}k}\quad{\mathbb{d}\zeta}}}}}} & (15)\end{matrix}$To improve computational efficiency, a partial literal evaluation of thetriple integration in Eq. (15) can be performed.

The literal analysis of Eq. (15) begins with a change of the order ofintegration to first evaluate the individual interference signalsg_(β,k,ζ,h) over all scan positions ζ at fixed β and k: $\begin{matrix}{q_{Κ,h} = {\int_{0}^{\infty}{\int_{0}^{1}{U_{\beta}V_{k}\beta\left\{ {\int_{- \infty}^{\infty}{g_{\beta,k,\zeta,h}\quad{\exp\left( {{\mathbb{i}}\quad Κ\quad\zeta} \right)}\quad{\mathbb{d}\zeta}}} \right\}\quad{\mathbb{d}\beta}\quad{{\mathbb{d}k}.}}}}} & (16)\end{matrix}$After expansion of the cosine term in g_(β,k,ζ,h) in the usual way using2 cos(u)=exp(iu)+exp(−iu),  (17)the inner integral over ζ evaluates to $\begin{matrix}\begin{matrix}{{\int_{- \infty}^{\infty}{g_{\beta,k,\zeta,h}\quad{\exp\left( {{\mathbb{i}}\quad Κ\quad\zeta} \right)}\quad{\mathbb{d}\zeta}}} = {{{\delta_{Κ}\left( {R_{\beta,k} + Z_{\beta,k}} \right)}\quad\ldots} +}} \\{\delta_{({Κ - {2\beta\quad{kn}_{0}}})}\sqrt{R_{\beta,k}Z_{\beta,k}}\quad{\exp\left\lbrack {{{\mathbb{i}}\quad 2\quad\beta\quad{kn}_{0}h} +} \right.}} \\{{\left. {{\mathbb{i}}\left( {\upsilon_{\beta,k} - \omega_{\beta,k}} \right)} \right\rbrack\quad\ldots} +} \\{\delta_{({Κ + {2\beta\quad{kn}_{0}}})}\sqrt{R_{\beta,k}Z_{\beta,k}}\quad{\exp\left\lbrack {{{- {\mathbb{i}}}\quad 2\quad\beta\quad{kn}_{0}h} -} \right.}} \\\left. {{\mathbb{i}}\left( {\upsilon_{\beta,k} - \omega_{\beta,k}} \right)} \right\rbrack\end{matrix} & (18)\end{matrix}$where we have used $\begin{matrix}{\delta_{Κ} = {\int_{- \infty}^{\infty}{{\exp\left( {{\mathbb{i}}\quad Κ\quad\zeta} \right)}\quad{\mathbb{d}\zeta}}}} & (19) \\{\delta_{({Κ \pm {2\beta\quad{kn}_{0}}})} = {\int_{- \infty}^{\infty}{{\exp\left( {{\mathbb{i}}\quad Κ\quad\zeta} \right)}\quad{\exp\left( {{\pm {\mathbb{i}}}\quad 2\quad\beta\quad{kn}_{0}\zeta} \right)}\quad{{\mathbb{d}\zeta}.}}}} & (20)\end{matrix}$The δ function carries with it the inverse physical units of theargument, in this case, an inverse wavenumber.

These delta functions validate an equivalency between the spatialfrequency K and the product 2βkn₀. A logical change of variables for thenext integration is thereforeβ={circumflex over (κ)}/2kn ₀  (21)dβ=d{circumflex over (κ)}/2kn ₀  (22)where {circumflex over (κ)} has the same meaning as the spatialfrequency K, but will be used as a free integration variable. Eq. (18)can be written $\begin{matrix}{q_{Κ,h} = {{\int_{0}^{\infty}{\int_{0}^{2{kn}_{0}}{{\delta_{Κ}\left( {R_{\hat{\kappa},k} + Z_{\hat{\kappa},k}} \right)}\quad\Gamma_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}\quad{\mathbb{d}k}\quad\ldots}}} +}} & (23) \\{\quad{\int_{0}^{\infty}{\int_{0}^{2{kn}_{0}}{\delta_{({Κ - \hat{\kappa}})}\sqrt{R_{\hat{\kappa},k}Z_{\hat{\kappa},k}}\quad{\exp\left\lbrack {{{\mathbb{i}}\hat{\kappa}h} +} \right.}}}}} & \quad \\{{\left. \quad{{\mathbb{i}}\left( {\upsilon_{\hat{\kappa},k} - \omega_{\hat{\kappa},k}} \right)} \right\rbrack\quad\Gamma_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}\quad{\mathbb{d}k}\quad\ldots} +} & \quad \\{\quad{\int_{0}^{\infty}{\int_{0}^{2{kn}_{0}}{\delta_{({Κ + \hat{\kappa}})}\sqrt{R_{\hat{\kappa},k}Z_{\hat{\kappa},k}}\quad{\exp\left\lbrack {{{- {\mathbb{i}}}\hat{\kappa}h} -} \right.}}}}} & \quad \\{\left. \quad{{\mathbb{i}}\left( {\upsilon_{\hat{\kappa},k} - \omega_{\hat{\kappa},k}} \right)} \right\rbrack\quad\Gamma_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}\quad{\mathbb{d}k}} & \quad \\{where} & \quad \\{\Gamma_{\hat{\kappa},k} = {\frac{U_{\hat{\kappa},k}V_{k}\hat{\kappa}}{4k^{2}n_{0}^{2}}.}} & (24)\end{matrix}$Note that by virtue of the change in variables, the β-dependence for theR, Z, υ, ω terms in Eq. (23) becomes a dependence upon {circumflex over(κ)} and k.

For the next step, we first note that $\begin{matrix}{{\int_{0}^{2{kn}_{0}}{\delta_{Κ}f_{\hat{\kappa},k}\quad{\mathbb{d}\hat{\kappa}}}} = {\delta_{Κ}\quad{\int_{0}^{\infty}{H_{({{2{kn}_{0}} - \hat{\kappa}})}f_{\hat{\kappa},k}\quad{\mathbb{d}\hat{\kappa}}}}}} & (25) \\{{\int_{0}^{2{kn}_{0}}{\delta_{({Κ - \hat{\kappa}})}f_{\hat{\kappa},k}\quad{\mathbb{d}\hat{\kappa}}}} = {f_{Κ,k}H_{Κ}H_{({{2{kn}_{0}} - Κ})}}} & (26) \\{{\int_{0}^{2{kn}_{0}}{\delta_{({Κ + \hat{\kappa}})}f_{\hat{\kappa},k}\quad{\mathbb{d}\hat{\kappa}}}} = {f_{{- Κ},k}H_{- Κ}H_{({{2{kn}_{0}} + Κ})}}} & (27)\end{matrix}$where H is the unitless Heaviside step function defined by$\begin{matrix}{H_{u} = \left\{ \begin{matrix}0 & {{{for}\quad u} < 0} \\1 & {otherwise}\end{matrix} \right.} & (28)\end{matrix}$and ƒ is an arbitrary function of K and k. Using Eqs. (25) through (27),Eq. (23) becomes $\begin{matrix}\begin{matrix}{q_{Κ,h} = {{\delta_{Κ}{\int_{0}^{\infty}{\int_{0}^{\infty}{{H_{({{2{kn}_{0}} - \hat{\kappa}})}\left( {R_{\hat{\kappa},k} + Z_{\hat{\kappa},k}} \right)}\quad\Gamma_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}\quad{\mathbb{d}k}\quad\ldots}}}} +}} \\{\int_{0}^{\infty}{H_{Κ}H_{({{2{kn}_{0}} - Κ})}\quad\sqrt{R_{Κ,k}Z_{Κ,k}}\quad{\exp\left\lbrack {{{\mathbb{i}}\quad Κ\quad h} +} \right.}}} \\{{\left. {{\mathbb{i}}\left( {\upsilon_{Κ,k} - \omega_{Κ,k}} \right)} \right\rbrack\quad\Gamma_{Κ,k}\quad{\mathbb{d}k}\quad\ldots} +} \\{\int_{0}^{\infty}{H_{- Κ}H_{({{2{kn}_{0}} + Κ})}\quad\sqrt{R_{{- Κ},k}Z_{{- Κ},k}}\quad{\exp\left\lbrack {{{\mathbb{i}}\quad Κ\quad h} -} \right.}}} \\{\left. {{\mathbb{i}}\left( {\upsilon_{Κ,k} - \omega_{{- Κ},k}} \right)} \right\rbrack\quad\Gamma_{{- Κ},k}\quad{\mathbb{d}k}}\end{matrix} & (29)\end{matrix}$Now using $\begin{matrix}{{\int_{0}^{\infty}{\int_{0}^{\infty}{H_{({{2{kn}_{0}} - \hat{\kappa}})}f_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}{\mathbb{d}k}}}} = {\int_{0}^{\infty}{\int_{0}^{\infty}{H_{({{2{kn}_{0}} - \hat{\kappa}})}f_{\hat{\kappa},k}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}}} & (30) \\{{\int_{0}^{\infty}{H_{K}H_{({{2{kn}_{0}} - K})}f_{K,k}{\mathbb{d}k}}} = {H_{K}{\int_{{K/2}\quad n_{0}}^{\infty}{f_{K,k}{\mathbb{d}k}}}}} & (31) \\{{\int_{0}^{\infty}{H_{- K}H_{({{2{kn}_{0}} + K})}f_{{- K},k}{\mathbb{d}k}}} = {H_{- K}{\int_{{{- K}/2}n_{0}}^{\infty}{f_{{- K},k}{\mathbb{d}k}}}}} & (32)\end{matrix}$we have the final result $\begin{matrix}{\begin{matrix}{q_{K,h} = {{\delta_{K}{\int_{\kappa = 0}^{\infty}{\int_{{\hat{\kappa}/2}n_{0}}^{\infty}{\left( {R_{\kappa,k} + Z_{\kappa,k}} \right)\Gamma_{\hat{\kappa},k}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}}} +}} \\{{H_{\kappa}{\exp\left( {{\mathbb{i}}\quad{Kh}} \right)}{\int_{{K/2}n_{0}}^{\infty}{\sqrt{R_{K,k}Z_{K,k}}{\exp\left\lbrack {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{K,k}} \right)} \right\rbrack}\Gamma_{K,k}{\mathbb{d}k}\quad\ldots}}} +} \\{H_{- K}{\exp\left( {{\mathbb{i}}\quad{Kh}} \right)}{\int_{{{- K}/2}n_{0}}^{\infty}{\sqrt{R_{{- K},k}Z_{{- K},k}}{\exp\left\lbrack {- {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{{- K},k}} \right)}} \right\rbrack}}}} \\{\Gamma_{{- K},k}{\mathbb{d}k}}\end{matrix}.} & (33)\end{matrix}$Because there are fewer integrations, Eq. (33) is significantly moreefficient computationally that the original triple integral of (15).

Some limit cases are interesting to solve analytically. For example, ifthe phase contribution (υ_(K,k)−ω_(K,k))=0 and the reflectivities R, Zare independent of incident angle and wavelength, then Eq. (33)simplifies to $\begin{matrix}\begin{matrix}{q_{K,h} = {{{\delta_{K}\left( {R + Z} \right)}{\int_{0}^{\infty}{\int_{{\hat{\kappa}/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}}} + {H_{K}{\exp\left( {{\mathbb{i}}\quad{Kh}} \right)}\sqrt{RZ}}}} \\{{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{K,k}{\mathbb{d}k}}} + {H_{- K}{\exp\left( {{\mathbb{i}}\quad{Kh}} \right)}\sqrt{RZ}{\int_{{{- K}/2}n_{0}}^{\infty}{\Gamma_{{- K},k}{\mathbb{d}k}}}}}\end{matrix} & (34)\end{matrix}$and we have only to handle integrals involving the weighting factor ΓK,kdefined in Eq. (24). This idealized case simplifies evaluation of twofurther limit cases for Eq. (34): Near-monochromatic illumination with ahigh-NA objective, and broadband illumination with low-NA.

For the case of a near-monochromatic light source having a narrowspectral bandwidth k_(Δ), we have the normalized spectrum$\begin{matrix}{V_{k} = {\frac{1}{k_{\Delta}}H_{({k - {k0}})}H_{({{k0} + k_{\Delta} - k})}}} & (35)\end{matrix}$where k0 is the nominal source wavenumber. The integrations in Eq. (34)are now of the form: $\begin{matrix}{{\int_{0}^{\infty}{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}} = {\frac{1}{4n_{0}^{2}k_{\Delta}}{\int_{0}^{\infty}{H_{({{k0} - {{\hat{\kappa}/2}n_{0}}})}\hat{\kappa}{\int_{k0}^{{k0} + k_{\Delta}}{\frac{U_{\hat{\kappa},k}}{k^{2}}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}}}}} & (36) \\{{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{K,k}{\mathbb{d}k}}} = {\frac{1}{4n_{0}^{2}k_{\Delta}}H_{({{k0} - {{K/2}n_{0}}})}K{\int_{k0}^{{k0} + k_{\Delta}}{\frac{U_{K,k}}{k^{2}}{{\mathbb{d}k}.}}}}} & (37)\end{matrix}$Assuming that U_(K,k) is essentially constant over the small bandwidthk_(Δ), we have $\begin{matrix}{{\int_{0}^{\infty}{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}} = {\int_{0}^{\infty}{H_{({{k0} - {{\hat{\kappa}/2}n_{0}}})}U_{\hat{\kappa},{k0}}\frac{\hat{\kappa}}{4n_{0}^{2}{k0}^{2}}{\mathbb{d}\hat{\kappa}}}}} & (38) \\{{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{K,k}{\mathbb{d}k}}} = {H_{({{k0} - {{K/2}n_{0}}})}U_{K,{k0}}{\frac{K}{4n_{0}^{2}{k0}^{2}}.}}} & (39)\end{matrix}$where in the evaluation of the integrals we have used $\begin{matrix}{{{\frac{- 1}{{k0} + k_{\Delta}} + \frac{1}{k0}} \approx \frac{k_{\Delta}}{k0}},} & (40)\end{matrix}$valid for a narrow bandwidth k_(Δ)□ k0. In particular, the positive,nonzero portion of the spectrum reduces to $\begin{matrix}{q_{{K > 0},h} = {\frac{H_{K}H_{({{k0} - {{K/2}n_{0}}})}U_{K,{2n_{0}{k0}}}K\sqrt{RZ}}{4n_{0}^{2}{k0}^{2}}{\exp\left( {{\mathbb{i}}\quad{Kh}} \right)}}} & (41)\end{matrix}$Consequently, for this special case of a narrow spectral bandwidth lightsource, constant reflectivities R, Z and no phase contributions{overscore (ω)},φ″_(K,h) =Kh.  (42)In this special case, the phase is linearly proportional to surfaceheight, consistent with conventional FDA. The spatial frequency also hasa direct correspondence to the directional cosine:K=β2n ₀ k0.  (43)Thus there is a one-to-one relationship between the spatial frequencycoordinate of the FDA spectra and the angle of incidence. Note furtherthe K weighting in the Fourier magnitude {square root}{square root over(Q_(K))} calculated from Eq. (41). This is evident in the examplespectrum FIG. 6(a), which shows the theoretical prediction for aperfectly uniform filling of the pupil plane over a range starting fromnormal incidence up to the directional cosine limit imposed by theobjective NA:β_(NA)={square root}{square root over (1−NA ²)}  (44)As a second example, consider the case of broadband illumination withuniform illumination restricted to a narrow range β_(Δ) of directionalcosines near normal incidence. The normalize pupil plane distribution isthen $\begin{matrix}{U_{\beta} = {\frac{1}{\beta_{\Delta}}H_{1 - \beta}{H_{\beta - {({1 - \beta_{\Delta}})}}.}}} & (45)\end{matrix}$After the change of variables, $\begin{matrix}{U_{K,k} = {\frac{1}{\beta_{\Delta}}H_{({{2{kn}_{0}} - K})}H_{\lbrack{K - {2{{kn}_{0}{({1 - \beta_{\Delta}})}}}}\rbrack}}} & (46)\end{matrix}$The definite integrals in Eq. (34) are in this case of the form$\begin{matrix}{{\int_{0}^{\infty}{\int_{{Κ/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}} = {\frac{1}{\beta_{\Delta}}{\int_{0}^{\infty}{\int_{{\hat{\kappa}/2}n_{0}}^{{\hat{\kappa}/{({1 - \beta_{\Delta}})}}2n_{0}}{\frac{V_{k}\quad\hat{\kappa}}{4k^{2}n_{0}^{2}}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}}}} & (47) \\{{\int_{{Κ/2}n_{0}}^{\infty}{\Gamma_{Κ,k}{\mathbb{d}k}}} = {\frac{1}{\beta_{\Delta}}{\int_{{Κ/2}n_{0}}^{{Κ/{({1 - \beta_{\Delta}})}}2n_{0}}{\frac{V_{k}\quad Κ}{4k^{2}n_{0}^{2}}{\mathbb{d}k}}}}} & (48)\end{matrix}$which evaluate to $\begin{matrix}{{\int_{0}^{\infty}{\int_{{Κ/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}} = {\int_{0}^{\infty}{\frac{V_{{\hat{\kappa}/2}n_{0}}}{2n_{0}}{\mathbb{d}\hat{\kappa}}}}} & (49) \\{{\int_{{Κ/2}n_{0}}^{\infty}{\Gamma_{Κ,k}{\mathbb{d}k}}} = {\frac{V_{{Κ/2}n_{0}}}{2n_{0}}.}} & (50)\end{matrix}$where we have used $\begin{matrix}{{\frac{\left( {1 - \beta_{\Delta}} \right)2n_{0}}{\hat{\kappa}} - \frac{2n_{0}}{\hat{\kappa}}} = {- \frac{2n_{0}\beta_{\Delta}}{\hat{\kappa}}}} & (51)\end{matrix}$The positive, nonzero portion of the spectrum is for this broadbandsource illumination and near-normal incidence is therefore$\begin{matrix}{q_{{Κ > 0},h} = {\frac{V_{{Κ/2}n_{0}}\sqrt{RZ}}{2n_{0}}\quad{\exp\left( {{\mathbb{i}}\quad Κ\quad h} \right)}}} & (52)\end{matrix}$This corresponds closely to the familiar result that the Fouriermagnitude {square root}{square root over (Q_(K))} is proportional to thesource spectrum distribution V_(K/2n) ₀ , as shown e.g. in FIG. 6(b) fora gaussian spectrum centered on a nominal or mean wavelength k0. Notethat Eq. (52) also conforms to the assumption of linear phase evolutionφ″_(K,h) =Kh  (53)consistent with conventional FDA.

Since the Fourier magnitude {square root}{square root over(Q_(K,h))}=|q_(K,h)| and phase φ″_(K,h)=arg(q_(K,h)) are derived fromthe Fourier Transform of the interference intensity I_(ζ,h), the inversetransform puts us back into the domain of real interference signals$\begin{matrix}{I_{\zeta,h} = {\int_{- \infty}^{\infty}{q_{\hat{\kappa},h}\quad{\exp\left( {{- {\mathbb{i}}}\quad\hat{\kappa}\zeta} \right)}\quad{\mathbb{d}\hat{\kappa}}}}} & (54)\end{matrix}$where once again we have used {circumflex over (κ)} as for the spatialfrequency to emphasize that it is a free variable of integration in Eq.(54). Thus one way to calculate the intensity signal is to generate theFourier components q_(K,h) by Eq. (33) and transform to I_(ζ,h) usingEq. (54).

We assume random polarization of the source light in the present model.This does not mean, however, that we should neglect polarizationeffects. Rather, in the above calculations, we assume an incoherentsuperposition of equally weighted results from the two orthogonalpolarization states s and p defined by the plane of incidence of theillumination. Using superscript notation for the polarizations,q _(β,k) =q _(β,k) ^(s) +q _(β,k) ^(p).  (55)Therefore, the average phase angle for unpolarized light at this β, kwould be

φ″_(β,k) =arg(q _(β,k) ^(s) +q _(β,k) ^(p)).  (56)Note that unless the magnitudes are identical for the two polarizationcontributions, most often

φ″_(β,k)≈(φ_(β,k) ^(s) +φ″ _(β,k) ^(p))/2.  (57)Also, unless q_(β,k) ^(s) and q_(β,k) ^(p) are perfectly parallel in thecomplex plane,

Q_(β,k)

≠(Q _(β,k) ^(s) +Q _(β,k) ^(p))/2.  (58)The same observation applies to the system and object reflectivitiesR_(β,k) ^(s), R_(β,k) ^(p) and Z_(β,k) ^(s), Z_(β,k) ^(p), respectively;they cannot be summed directly unless they have identical phases.

Provided that we take proper care of the polarization effects in thecalculation of the object surface reflectivity, the modeling remainsfairly straightforward and is flexible enough to handle the moreinteresting cases of polarized light further down the line.

The next step is to translate to discrete numerical formulas, in view ofa software development. We redefine the relationship between theinterference signal I_(ζ,h) and the Fourier spectrum q_(K,h) usingdiscrete Fourier transforms as $\begin{matrix}{q_{Κ,h} = {\frac{1}{\sqrt{N}}{\sum\limits_{\zeta}{I_{\zeta,h}\quad\exp\quad\left( {{\mathbb{i}}\quad Κ\quad\zeta} \right)}}}} & (59) \\{I_{\zeta,h} = {\frac{1}{\sqrt{N}}\left\lbrack {q_{0} + {\sum\limits_{Κ > 0}{q_{Κ,h}\quad{\exp\left( {{- {\mathbb{i}}}\quad Κ\quad\zeta} \right)}}} + {\sum\limits_{Κ > 0}{{\overset{\_}{q}}_{Κ,h}\quad{\exp\left( {{\mathbb{i}}\quad Κ\quad\zeta} \right)}}}} \right\rbrack}} & (60)\end{matrix}$where {overscore (q)}_({circumflex over (κ)},h) is the complex conjugateof q_({circumflex over (κ)},h) and there are N discrete samples in theinterference signal I_(ζ,h). In Eq. (60) and what follows, we have setaside the use of a free variable {circumflex over (κ)} that wasimportant in the derivations but it is no longer needed as a substitutefor the spatial frequency K. The predicted positive-frequency FDAcomplex spectrum is then $\begin{matrix}{q_{{Κ \geq 0},h} = {\rho_{Κ \geq 0}\quad{\exp\left( {{\mathbb{i}}\quad Κ\quad h} \right)}}} & (61)\end{matrix}$where the normalized, height-independent coefficients are$\begin{matrix}{\rho_{Κ > 0} = {\frac{\sqrt{N}}{\Upsilon}{\sum\limits_{k}{H_{k - {{Κ/2}n_{0}}}\sqrt{R_{Κ,k}Z_{Κ,k}}\quad{\exp\left\lbrack {{\mathbb{i}}\left( {\upsilon_{Κ,k} - \omega_{Κ,k}} \right)} \right\rbrack}\quad\Gamma_{Κ,k}}}}} & (62) \\{\rho_{0} = {\frac{\sqrt{N}}{\Upsilon}{\sum\limits_{Κ \geq 0}{\sum\limits_{k}{H_{k - {{Κ/2}n_{0}}}\quad\left( {R_{Κ,k} + Z_{Κ,k}} \right)\quad\Gamma_{Κ,k}}}}}} & (63)\end{matrix}$where the normalization for the range of integration is $\begin{matrix}{\Upsilon = {\sum\limits_{Κ \geq 0}{\sum\limits_{k}{H_{k - {{K/2}n_{0}}}\Gamma_{Κ,k}}}}} & (64)\end{matrix}$The Heaviside step functions H in Eq. (62) prevent unnecessarycontributions to the sums. The weighting factor Γ_(K,k) is as defined inEq. (24).

To compare experiment with theory, we use Eq. (61) to generate anexperimental FDA spectrum and Eq. (62) to transform back into the spacedomain for the theoretical prediction of I_(ζ,h). This is mostefficiently performed by fast Fourier transforms (FFT). The propertiesof the FFT determine the range of K values. If the N discrete samplesfor I_(ζ,h) are spaced by an increment ζ_(step), there will be N/2+1positive spatial frequencies starting from zero and rising to N/2 cyclesper data trace, spaced by an increment $\begin{matrix}{Κ_{step} = {\frac{2\quad\pi}{N\quad\zeta_{step}}.}} & (65)\end{matrix}$To facilitate phase unwrapping in the frequency domain, we try to adjustthe zero position for the scan so that it is near the signal peak, thusreducing the phase slope in the frequency domain. Because an FFT alwaysassumes that the first data point in the scan is at zero, the signalshould be properly offset.

We now focusing on modeling a sample surface with a thin film.

FIG. 7 shows two surface types, with and without a thin film. For bothcases, we define an effective amplitude reflectivity z_(β,k) accordingtoz _(β,k)={square root}{square root over (Z _(β,k))} exp(iω _(β,k))  (66)where Z_(β,k) is the intensity reflectivity and ω_(β,k) is the phasechange on reflection. The subscripts β,k emphasize a dependency on thedirectional cosine of the illuminationβ₀=cos(ψ₀),  (67)where ψ₀ is the incident angle, and on the wavenumberk=(2π/λ)  (68)where λ is the wavelength of the light source. The subscript β will beunderstood to refer to the first incident directional cosine β₀.

The surfaces are characterized in part by their index of refraction. Theindex of the surrounding medium, usually air, is n₀. For the simplesurface FIG. 7(a) there is only one index n₁. For the thin film in FIG.7(b), there are two surface indices, n₁ for the transparent or partiallytransparent film and n₂ for the substrate. Most generally, theserefractive indices are complex numbers characterized by a real part andan imaginary part. For example, a typical index, e.g. for chrome atλ=550 nm, is n₁=3.18+4.41i, where we are adopting the convention wherethe imaginary part is defined as positive.

The index of refraction of a material depends on the wavelength. Thedispersion in refractive index n₀ for the air is not very significant,but is important for many sample surfaces, especially metals. Over smallwavelength changes near a nominal k0, most materials have a nearlylinear dependence on wavenumber, so that we can write $\begin{matrix}{n_{1,k} = {v_{1}^{(0)} + {k\quad v_{1}^{(1)}}}} & (69)\end{matrix}$where v₁ ⁽⁰⁾, v₁ ⁽¹⁾ are the intercept and slope, respectively, for theindex of refraction n₁ at the nominal wavenumber k0.

The most common use of the refractive index is Snell's law. Referring toFIG. 7(b), the refracted beam angle internal to the film is$\begin{matrix}{{\psi_{1,\beta,k} = {\arcsin\left\lbrack {\frac{n_{0}}{n_{1,\beta,k}}{\sin\left( \psi_{0} \right)}} \right\rbrack}},} & (70)\end{matrix}$where ψ₀ is the angle within the medium of index n₀ incident on the topsurface of the medium of index n₁, and ψ_(1,β,k) is the angle ofrefraction. It is possible for these angles to take on complex values ifthe indices are complex, indicating partially evanescent propagation.

The complex amplitude reflectivity of a boundary between two mediadepends on the polarization, the wavelength, the angle of incidence andthe index of refraction. The s- and p-polarization reflectivities of thetop surface of the film in FIG. 7(b) are given by the Fresnel formulae$\begin{matrix}{\vartheta_{1,\beta,k}^{s} = \frac{\tan\left( {\psi_{0} - \psi_{1,\beta,k}} \right)}{\tan\left( {\psi_{0} + \psi_{1,\beta,k}} \right)}} & (71) \\{\vartheta_{1,\beta,k}^{p} = \frac{\tan\left( {\psi_{0} - \psi_{1,\beta,k}} \right)}{\tan\left( {\psi_{0} + \psi_{1,\beta,k}} \right)}} & (72)\end{matrix}$The dependence on β,k results from the angles ψ₀, ψ_(1,β,k), the exitangle ψ_(1,β,k) introducing a k dependency via the refractive indexn_(1,k). Similarly, the substrate-film interface reflectivities are$\begin{matrix}{\vartheta_{2,\beta,k}^{p} = \frac{\tan\left( {\psi_{1,\beta,k} - \psi_{2,\beta,k}} \right)}{\tan\left( {\psi_{1,\beta,k} + \psi_{2,\beta,k}} \right)}} & (73) \\{\vartheta_{2,\beta,k}^{s} = {- \frac{\tan\left( {\psi_{1,\beta,k} - \psi_{2,\beta,k}} \right)}{\tan\left( {\psi_{1,\beta,k} + \psi_{2,\beta,k}} \right)}}} & (74)\end{matrix}$Note that in the Fresnel equations, if the angle of incidence andrefraction are the same, the reflectivity for both polarizations goes tozero.

For a simple surface (no thin film), the sample surface reflectivity isidentical to the top-surface reflectivityz_(β,k)=

_(1,β,k) (simple surface, no thin film)  (75)Consequently, the phase change on reflection (PCOR) caused by thesurface reflection isω_(β,k) =arg(

_(1,β,k)).  (76)Note that to satisfy the boundary conditions, the s-polarization “flips”upon reflection (=π phase shift for a dielectric), whereas thep-polarization does not. The distinction between polarization statesbecomes meaningless exactly at normal incidence, which in any caseresults in a division by zero in the Fresnel equations and a differentformula handles this limit case.

When using the plus sign convention for the complex part of the index ofrefraction, the greater the absorption (complex part), the greater thePCOR ω_(β,k). In other words, a larger absorption coefficient isequivalent to a decrease in effective surface height. This makesintuitive sense—one imagines absorption as a penetration of the lightbeam into the material prior to reflection, rather than a cleanreflection and transmission right at the boundary. Following our usualconvention, for which an increase in surface height corresponds to apositive change in the phase difference between the reference andmeasurement surfaces, a positive surface PCOR subtracts from theinterferometer phase.

A thin film is a special case of a parallel plate reflection. The lightpasses through the top surface partially reflected (see FIG. 7) andcontinues to the substrate surface where there is a second reflectionwith a phase delay with respect to the first. However, this is not theend of the story. The light reflected from the substrate is once againpartially reflected when passing back up through the top surface,resulting in an additional reflected beam heading south again to thesubstrate. This goes on in principle forever, with each additionalreflection a little weaker than the last. Assuming that all of thesemultiple reflections survive to contribute to the final surfacereflectivity, the infinite series evaluates to $\begin{matrix}{z_{\beta,k} = \frac{\vartheta_{1,\beta,k} + {\vartheta_{2,\beta,k}{\exp\left( {2{\mathbb{i}}\quad{kL}\quad\beta_{1,\beta,k}n_{1,k}} \right)}}}{1 + {\vartheta_{1,\beta,k}\vartheta_{2,\beta,k}{\exp\left( {2{\mathbb{i}}\quad{kL}\quad\beta_{1,\beta,k}n_{1,k}} \right)}}}} & (77)\end{matrix}$  β_(1,β,k)=cos(ψ_(1,β,k)).  (78)As a note of clarification, recall the β dependency of β_(1,β,k) refersto a dependency on the incident directional cosine β₀ in the ambientmedium of index n₀. The same Eq. (77) applies to both polarizationstates, with corresponding single-surface reflectivities.

Inspection of these equations shows why conventional FDA processingbreaks down in the presence of thin films. Conventional FDA determinessurface height by a linear fit to the Fourier phase spectrum weighted bythe Fourier power spectrum, using broadband (white) light to generatethe Fourier spatial frequency spread. The idea is that the phaseevolution comes from the expected linear phase dependence on surfaceheight. Any other constant offset or linear coefficients (e.g.,“dispersion”) associated with the surface characteristics are removed bysystem characterization or by simply ignoring those phase contributionsthat do not change with field position.

This works perfectly fine for simple surfaces. With unpolarized light,and most likely with the circularly-polarized light, the wavelengthdependence of the PCOR is nearly linear with respect to wavenumber andconstant for a given material. In the presence of a thin film, however,the conventional analysis breaks down. The phase becomes nonlinear andthe phase slope becomes sensitive to film thickness, which may bevarying across the field of view. Therefore, the present analysisdetermines key parameters of the surface structure such as filmthickness by comparing experimental data to a theoretical prediction,using our knowledge of how e.g. a thin film modulates the reflectivityof the surface.

We now discuss how comparison of experimental data to a library oftheoretical predictions provides surface structure parameters such asfilm thickness and phase change on reflection (PCOR). In the case of afilm of unknown thickness, the library for a single surface type, e.g.SiO₂ on Si, would range over many possible film thicknesses. Infrequency domain embodiments, the idea is to search this library for amatch to those characteristics of the FDA spectra that are independentof surface topography, for example, a distinctive structure to themagnitude spectrum resulting from a thin-film interference effect. Thecomputer then uses the library spectrum to compensate the FDA data,allowing for a more accurate surface topography map.

In one embodiment, the library contains example FDA spectra for surfacestructures, each spectrum providing a series of complex coefficientsρ_(K) representing Fourier coefficients as a function of spatialfrequency K. These spectra are the Fourier transforms of intensity dataI_(ζ,h) acquired during a scan ζ of the optical path length of aninterferometer. The spatial frequency K is proportional to the angularwavenumber k=2π/λ for a segment of the source light spectrum, the indexof refraction n₀ of the ambient medium, and the directional cosineβ=cos(ψ), where ψ is the angle of incidence for a ray bundle directed tothe object surface:K=2βkn ₀.  (79)The ρ_(K) coefficients for the prediction library include the opticalproperties of the surface that can influence the appearance of the FDAspectra, with the exception of surface height.

Predicting the FDA spectra involves an integral representing theincoherent sum of ray bundles over a range of incident angles ψ andangular wavenumbers k for the source light. As described above, thenumerical integration can reduce to a computationally-efficient singlesum over N angular wavenumbers k, weighted by a factor Γ_(K,k):$\begin{matrix}{\rho_{K > 0} = {\frac{\sqrt{N}}{\Upsilon}{\sum\limits_{k}{H_{k - {{K/2}n_{0}}}\sqrt{R_{K,k}Z_{K,k}}{\exp\left\lbrack {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{K,k}} \right)} \right\rbrack}\Gamma_{K,k}}}}} & (80) \\{\rho_{0} = {\frac{\sqrt{N}}{\Upsilon}{\sum\limits_{K \geq 0}{\sum\limits_{k}{{H_{k - {{K/2}n_{0}}}\left( {R_{K,k} + Z_{K,k}} \right)}\Gamma_{K,k}}}}}} & (81)\end{matrix}$The weighting factor is $\begin{matrix}{{\Gamma_{K,k} = \frac{K\quad U_{K,k}V_{k}}{4k^{2}n_{0}^{2}}},} & (82)\end{matrix}$where V_(k) is the source spectrum and U_(K,k) is the pupil-plane lightdistribution. The corresponding normalization Υ is the sum over allspatial frequencies of the weighting factor $\begin{matrix}{\Upsilon = {\sum\limits_{K \geq 0}{\sum\limits_{k}{H_{k - {{K/2}n_{0}}}\Gamma_{K,k}}}}} & (83)\end{matrix}$where Υ is a normalization to be defined shortly and H is the Heavisidestep function.

The distinctive characteristics of an object surface structure,particularly of a thin film, enter into the spectrum ρ_(K) through theobject-path phase ω_(K,k) and reflectivity Z_(K,k), as detailed above.Equally important are the reference-path phase υ_(K,k) and reflectivityR_(K,k), which depend on the scanning interferometer itself. Suchfactors can be determined by theoretically modeling the scanninginterferometer or by calibrating it with a test sample having knownproperties, as described further below.

The typical prediction library for a thin film is a series of spectraρ_(K) indexed by film thickness L. The stored spectra cover only anarrow spatial frequency region of interest (ROI), usually 15 or 16values for a 256-frame intensity data acquisition, the remainder of thevalues outside this ROI being zero. The limits of the ROI follow fromthe definition of the spatial frequency:K ^(min)=2β^(min) k ^(min) n ₀  (84)K ^(max)=2β^(max) k ^(max) n ₀  (85)A typical range of spatial frequencies for a scanning interferometerbased on a 100×Mirau objective and a narrow bandwidth, 500-nm lightsource is 2.7 μm⁻¹ to 4.0 μm⁻¹. For computational efficient, a denselook up table, indexed by 0.5 to 5 nm between sample spectra, can beused rather than an analytical search routine that involvesrecalculation using Eqs. (80)-(83) several times for each pixel.

The library search involves the following steps: (1) Select a predictedFDA spectrum from the library corresponding to a specific surface type,(2) calculate how closely this spectrum matches the experimental datausing a merit function, then (3) repeat through several or all of thelibrary data sets to determine which theoretical spectrum provides thebest match. What we are looking for is a “signature” in the frequencydomain that relates uniquely to surface characteristics such as thinfilms, dissimilar materials, step structures, roughness, and theirinteraction with the optical system of the interferometer. Thiscomparison therefore explicitly filters away the linear rate of changeof phase with spatial frequency, which is the one characteristic of theFDA spectrum that varies directly with surface topography and istherefore irrelevant to the library search.

In comparing spectra, there is a benefit to separating the phase andmagnitude contributions to the merit calculation. Thus for the theory,we haveP_(K)=|ρ_(K)|  (86)φ_(K)=connect_(K) [arg(ρ_(K))],  (87)where connect_(K) is a function that removes 2-π steps in the spatialfrequency dependence of φ_(K,h). For the experimental data we have$\begin{matrix}{P_{K}^{ex} = {q_{K,h}^{ex}}} & (88) \\{{\phi_{K,h}^{n_{ex}} = {{connect}_{K}\left\lbrack {\arg\left( q_{K,h}^{ex} \right)} \right\rbrack}},} & (89)\end{matrix}$The double prime for φ″_(K) ^(ex) indicates an uncertainty in the fringeorder from both pixel to pixel and overall with respect to the startingpoint in the scan. The experimental data necessarily include a slopeterm related to the local surface height; this is the reason for the useof the q symbol instead of the p symbol.

For a specific set of trial surface parameters, we can calculate a phasedifference $\begin{matrix}{_{K,h}^{n} = {\phi_{K,h}^{n_{ex}} - \phi_{K}}} & (90)\end{matrix}$The phase difference ζ″_(K,h) is the compensated FDA phase, assumingthat the trial parameters are correct. A good match of theory toexperiment yields a phase ζ″_(K,h) that in principle is a simple linearfunction of spatial frequency K with an intercept of zero (i.e., zerophase gap). Thus, looking ahead, the successfully compensated phaseζ″_(K,h) is what we shall eventually feed downstream to a conventionalFDA analysis, which assumes that the slope of the phase in frequencyspace is directly proportional to surface height.

Based on the observations of the previous paragraph, there are twofeatures of interest in the compensated phase ζ″_(K,h) that allow us toevaluate the match of theory to experiment independent of surfaceheight. The first is the phase gap A″ or K=0 intercept value ζ″_(K=0,h)obtained by a linear fit, and the second is the residual nonlinearitywith respect to wavenumber after a linear fit. Corresponding meritfunctions are, for example, $\begin{matrix}{\chi_{\phi} = \left\lbrack {\frac{A^{''}}{2\quad\pi} - {{round}\left( \frac{A^{''}}{2\quad\pi} \right)}} \right\rbrack^{2}} & (91) \\{\chi_{\phi\quad{non}} = \frac{\sum\limits_{k > 0}{\left( {\varsigma_{K,h}^{''} - {\sigma_{h}K} - A^{''}} \right)^{2}P_{K,h}^{ex}}}{\sum\limits_{K > 0}P_{K,h}^{ex}}} & (92)\end{matrix}$where σ_(h) is the slope of the (magnitude weighted) linear fit to thecompensated phase ζ″_(K,h). The round( ) function in Eq. (91) limits thephase gap A″ to the range ±π.

Although a library search can proceed using phase information alone,i.e. by minimizing one or both of the merit function values χ_(φ) and/orχ_(φnon), we also have important and useful signatures in the Fouriermagnitude. The magnitude is particularly interesting in that it isinherently independent of surface height. Thus for example, we candefine in approximate analogy with the phase merits the followingmagnitude merit functions: $\begin{matrix}{\chi_{P} = \left\lbrack \frac{\sum\limits_{K > 0}\left( {P_{K,h}^{ex} - P_{K,h}} \right)}{\sum\limits_{K > 0}\left( {P_{K,h}^{ex} + P_{K,h}} \right)} \right\rbrack^{2}} & (93) \\{\chi_{Pnon} = \frac{\sum\limits_{K > 0}\left( {{\Omega^{- 1}P_{K,h}^{ex}} - P_{K,h}} \right)^{2}}{\sum\limits_{K > 0}\left( {{\Omega^{- 1}P_{K,h}^{ex}} + P_{K,h}} \right)^{2}}} & (94)\end{matrix}$where Ω is the empirical scaling factor $\begin{matrix}{\Omega = {\sum\limits_{K > 0}{P_{K,h}^{ex}/{\sum\limits_{K > 0}{P_{K,h}.}}}}} & (95)\end{matrix}$The merit χ_(P) is most closely related to the overall reflectivity ofthe object surface, independent of spatial-frequency dependence, whereasχ_(Pnon) expresses how well the theoretical and experimental magnitudeplots match in shape.

The magnitude merit functions χ_(P) and/or χ_(Pnon) are in addition toor even in place of the phase merits χ_(φ) and/or χ_(φnon). A generallibrary search merit function is therefore $\begin{matrix}{\chi = {{w_{\phi}\chi_{\phi}} + {w_{\phi\quad{non}}\chi_{\phi\quad{non}}} + {w_{P}\chi_{P}} + {w_{P\quad{non}}\chi_{P\quad{non}}}}} & (96)\end{matrix}$where the w are weighting factors. In principle, one can determine theweights in Eq. (96) knowing the standard deviation for the variousparameters. A more empirical approach is to try out various weights onreal and simulated data and see how well they work. For the examplesthat follow, we select equal weights w_(φ)=w_(φnon)=w_(P)=w_(Pnon)=1 forall merit contributions.

The examples in FIGS. 8-13 illustrate the merit-function searchprocedure for six SiO₂ on Si film thicknesses: 0, 50, 100, 300, 600, and1200 nm, respectively. A single library for all examples encompasses therange from 0 to 1500 nm in 2-nm intervals. The data are simulations,free of noise. As in all the examples described herein, the scan step is40 mm, the source wavelength is 498 mm, and the source gaussian FWHM is30 nm (quasi-monochromatic).

The most interesting aspect of these simulated searches is the behaviorof the four merit functions. Generally, we observe that inclusion ofthese four functions helps to reduce the ambiguity in the final meritvalues, there being a strong periodicity for individual merit values asa function of film thickness. Another general observation is that themerits based on nonlinearity, both in phase and magnitude, are mosteffective at 300 nm and above, whereas the phase gap and averagemagnitude are dominant below 300 run film thickness. This shows that theχ_(φ), χ_(P) merit functions are especially useful to the really thinfilms, which places importance on system characterization, which couplesdirectly into the phase gap and magnitude results.

Once we determine the thin film thickness (or identify the material orother uses for the algorithm), FDA processing proceeds in the usual way,using however the corrected FDA phase ζ″_(K,h) instead of the originalexperimental phase data. In principle, if the modeling has beensuccessful, ζ″_(K,h) should be free of nonlinearities and the phase gapshould be zero. The next step therefore is a linear fit to the phasespectrum ζ″_(K,h). It appears more effective for high-NA FDA to use themagnitude spectrum P_(K) in place of magnitude squared. The fit providesfor each pixel a slopeσ_(h) ≈dζ″ _(K,h) /dK  (97)and an intercept (phase gap)A″≈ζ″_(K=0,h).  (98)Note that the phase gap A″ carries the double prime inherited from thefringe order uncertainty in the phase data. The slope σ_(h) is free ofthis uncertainty. From the intercept A″ and the slope σ_(h), we definefor a specific mean or nominal spatial frequency K0 a “coherenceprofile”Θ_(h)=σ_(h) K0  (99)and a “phase profile”θ″_(h)=Θ_(h) +A″.  (100)We then removes the pixel-to-pixel fringe order uncertainty in the phaseθ″_(h): $\begin{matrix}{\theta^{\prime} = {\theta^{''} - {2\quad\pi\quad{{round}\quad\left\lbrack \frac{A^{''} - \alpha^{\prime}}{2\quad\pi} \right\rbrack}}}} & (101)\end{matrix}$where α′ is an approximation to the original phase gap A″ that is freeof pixel-to-pixel 2π steps.

Finally, the height profile follows fromh′=θ′/K0.  (102)Note that it is not necessary to subtract the phase offset γ, because ithas already been done in generating the compensated phases ζ_(K,h).

The first example of a surface topography measurement (FIG. 14) is apure simulation. The surface topography is everywhere zero, but there isan underlying film layer that progresses from 0 to 1500 nm in 10 nmincrements. Using the same prediction library as in FIGS. 8-13, thistest demonstrates unambiguous determination of film thickness throughoutthe range of the prediction library, albeit for perfect, noise-freedata.

The next example (FIG. 15) is also a simulation, but with additivenoise. The random additive noise is gaussian, with a standard deviationof 2 bits out of an average 128 intensity bits, which looks to betypical of real data. The results are clearly satisfactory despite thesignificant difference in reflectivity between SiO₂ and Si (4% to 45%).

We now address system characterization.

We define a phase offset γ_(sys) and a linear dispersion τ_(sys) usingdata collected during a system characterization procedure. To includesystem characterization data, we correct the Fourier-transformedexperimental data q_(K) ^(ex) prior to the library search and prior toany other FDA processing on a pixel-by-pixel basis usingq _(K>0) ^(ex) =M ⁻¹ exp [−iγ _(sys) −i(K−K0)τ_(sys) ]q _(K>0)^(ex).  (103)where K0 is the nominal spatial frequency, which represents the nominalspectral frequency for the FDA data set, as identified e.g. by locatingthe midpoint of the ROI. Note that the theoretical library remainsunchanged. The scaling coefficient M (greek capital “M”) is a new systemcharacterization that makes it possible to use object surfacereflectivity as a parameter in the library search.

The phase offset γ_(sys) and the system phase gap A_(sys) as a functionsof field position can be stored as a function of field position, andcalculate the true system dispersion according toτ_(sys)=(γ_(sys) −A _(sys))/K0.  (104)The magnitude coefficient M is also field dependent.

The creation of system characterization data proceeds in a mannersimilar to that described above for the object sample. We move to anartifact having known characteristics, measure it, and determine thesystem characterization by looking at how the results differ from whatwe would expect for a perfect system. Specifically, using a known samplefor which the correct library entry is predetermined, we generate thephase gap A″ as in Eq. (98) and a final height h′ as in Eq. (102). Then,assuming a perfectly flat artifact, we calculate the system phase offsetΓ_(sys) =K0h′  (105)and the system phase gapA _(sys)=connect_(xy)(A″)  (106)where connect_(xy)( ) is pixel-to-pixel phase unwrapping. The magnitudemap is $\begin{matrix}{M_{sys} = {\sum\limits_{K > 0}{P_{K,h}^{ex}/{\sum\limits_{K > 0}{P_{K,h}.}}}}} & (107)\end{matrix}$In some embodiments, several system characterizations can be averaged,perhaps using artifacts having similar surface structure to the finalapplication (e.g. SiO2 on Si) over a range of sample types.

In much of the description and simulations above we have focused on thinfilm surface structures, however, the analysis is also applicable toother types of complex surface structures. In what follows we show howthe scanning interferometry data can be analyzed to account for surfacestructures that are smaller than the optical resolution of the scanninginterferometer microscope. The optical resolution is ultimately limitedby the wavelength of the light source and the NA of the light collectionoptics.

FIG. 16 a shows height profiles determined from actual scanninginterferometry data of a 2400 lines per mm (lpmm) grating having apeak-to-valley (PV) modulation depth of 120 nm using a light source at a500-nm nominal wavelength. The top profile in FIG. 16 a shows the heightprofile determined using a conventional FDA analysis. The conventionalanalysis indicates a PV modulation depth of only about 10 nm, greatlyunderestimating the actual modulation depth. This inaccuracy occursbecause the grating has features at the limit of the optical resolutionof the 500-nm instrument. This is so even though the pixel resolution ofthe camera in the instrument is more than sufficient to accuratelyresolve the grating.

One way of thinking about this effect is that the scanninginterferometry signal for a first camera pixel generally correspondingto a first surface location also includes contributions from adjacentsurface locations when those additional surface locations have surfacefeatures sufficiently sharp relative to the light wavelength to diffractlight to the first pixel. The surface height features from thoseadjacent surface locations corrupt conventional analysis of the scanninginterferometry signal corresponding to the first surface location.

At the same time, however, this means that the scanning interferometrysignal corresponding to the first surface location includes informationabout the complex surface features nearby. FIG. 17 illustrates this byshowing the scanning interferometry signal from pixels corresponding tovarious locations about a step height feature. For the signal in (a) thestep height is to the right of the pixel and higher, for the signal in(b) the step passes directly through the pixel, and for the signal in(c) the step is to the left of the pixel and is lower. One signaturethat is immediately apparent in the signals is the reduction in fringecontrast in (b) relative to (a) and (c). For example, if the step heightwas equal to one-quarter of the wavelength and the pixel locationcorresponded exactly to the position of the step height, the fringecontrast in (b) should disappear entirely because interference from thetwo sides of the step would exactly cancel one another. There is alsomuch information in the signals in shown in (a) and (c). For example,FIG. 18 shows the nonlinear distortions in the frequency domain phasespectra for the signals (a) and (c) of FIG. 17, respectively, resultingfrom the nearby step height. These spectra are indicated as (a) and (b),respectively, in FIG. 18. In the absence of the step height, thefrequency domain phase spectra would be linear. Thus, the nonlinearfeatures in the frequency domain phase spectrum for pixels correspondingto surface locations adjacent to the step height nonetheless includeinformation about the step height.

To more accurately measure the surface profile of a test surface in thepresence of such under-resolved surface features, we can use the librarysearching technique described above for thin films. For example, for thecase of a test surface with an under-resolved grating, a series of modelFDA spectra are generated for different values of the PV modulationdepth and offset position. As in the thin film examples, the surfaceheight for the model spectra remains fixed. The analysis then continuesas in the thin film examples above, except that rather than the modelspectra being parameterized by thin film thickness, they areparameterized by modulation depth and offset position. Comparisonbetween signatures of the FDA spectra for the actual test surface andthe different model spectra can then be used to determine a match. Basedon the match, distortions in the actual FDA spectrum for each pixelcaused by the presence of the grating are removed so that the surfaceheight for each pixel can be determined using conventional processing.The results of such an analysis using the same merit functions asdescribed above for the thin films are shown in FIGS. 16 b and 19 b.

FIG. 16 b shows the height profile determined using the library searchanalysis for 2400 lines per mm grating described above with reference toFIG. 16 a. The same data was used in the FIGS. 16 a and 16 b, however,the library search analysis determined the PV modulation depth for thegrating to be 100 nm, much closer to the actual 120-nm modulation depththan the 10-nm result determined by conventional FDA processing in FIG.16 a. FIGS. 19 a and 19 b show a similar analysis for a simulation witha discrete step height and assuming a nominal 500-nm light source. FIG.19 a shows the height profile determined using conventional FDAprocessing (solid line) compared to the actual height profile for thesimulation (dotted line). FIG. 19 b shows the height profile determinedusing the library search method (solid line) compared to the actualheight profile for the simulation (dotted line). The parameters for themodel spectra in the library search were location and step heightmagnitude. As illustrated, the library search analysis improves lateralresolution from about 0.5 microns to about 0.3 microns.

In the detailed analyses described above the comparison betweeninformation in the actual data and information corresponding to thedifferent models has occurred in the frequency domain. In otherembodiments, the comparison can be made in the scan coordinate domain.For example, while changes in the absolute position of the fringecontrast envelope is generally indicative of changes in surface heightat a first surface location corresponding to the signal in question, theshape of the signal (independent of its absolute position) containsinformation of complex surface structure, such as underlying layers atthe first surface location and/or surface structure at adjacentlocations.

One simple case is to consider to the magnitude of the fringe contrastenvelope itself. For example, when a thin film thickness is very smallrelative to the range of wavelengths produced by the light source, theinterference effects produced by the thin film become wavelengthindependent, in which case thin film thickness directly modulates themagnitude of the fringe contrast envelope. So, in general, the fringecontrast magnitude can be compared to that for models corresponding todifferent thin film thicknesses to a identify a match for a particularthin film thickness (taking into account systematic contributions fromthe interferometer itself).

Another simple case is to look at the relative spacings of the zerocrossings of the fringes under the fringe contrast envelope. For asimple surface structure illuminated with a a symmetric frequencydistribution, the relative spacings between the different zero crossingsshould be nominally the same. Variations in the relative spacings aretherefore indicative of complex surface structure (when taking intoaccount systematic contributions from the interferometer itself) and cancompared to models for different complex surface structures to identifya match to a particular surface structure.

Another case is to perform a correlation between the scan-domain signaland the scan-domain signals corresponding to different models of thetest surface. A match generally corresponds to the correlation that hasthe highest peak value, which indicate the model whose scan-domainsignal has a shape most similar to the shape of the actual signal. Notethat such analysis is generally independent of surface height because adifference between the surface height of the actual sample and that ofeach model only shifts the location of peak in the correlation function,but does not effect, in general, the peak value itself. On the otherhand, once the correct model is identified, the location of the peak inthe correlation function of the correct model yields the surface heightfor the test sample, without the need for further analysis (such asconventional FDA).

Like the analysis in the spatial frequency domain, an analysis in thescan-coordinate domain can be used for many different types of complexsurfaces, including not only thin films, but also other complex surfacestructures such as under-resolved surface height features as describedabove.

We now describe in detail a scan-coordinate library search analysis theinvolves a correlation between the signal for the test sample andcorresponding signals for various models of the test sample.

The approach sets aside any assumptions about the interference patternother than to say that all pixels in a data set corresponding to surfacelocations with the same complex surface characteristics contain the samebasic, localized interference pattern, only shifted in position (andpossibly rescaled) for each pixel. It does not matter what the signalactually looks like, whether it is a gaussian envelope or has a linearphase behavior in the frequency domain or whatever. The idea is togenerate a sample signal or template that represents this localizedinterference pattern for different models of complex surface structuresfor the test object, and then for each pixel, find the model whoselocalized interference pattern best matches the shape of the actuallocalized interference pattern, and for that model, find the scanposition within the data set that provides the best match between theinterference pattern template and the observed signal—which gives thesurface height. Several techniques are available for pattern matching.One approach is to mathematically correlate each template with the data.Using a complex (i.e. real plus imaginary) template function for eachmodel, we recover two profiles, one closely associated with the envelopeof the signal and the other associated with the phase of the underlyingcarrier signal.

In one embodiment, for example, the analysis for each pixel would beinclude: (1) selecting a test template from a library of templatescalculated or recorded for a specific value of an adjustable parameter,such as film thickness; (2) finding the local surface height using theselected test template and a correlation technique (an example of whichis described below); (3) recording the peak merit function value for theselected test template based on the correlation technique; (4) repeatingsteps 1-3 for all or a subset of the templates in the library; (5)determining which test template provides the best match (=highest peakmerit function value); (6) recording the value for the adjustableparameter for the best-matched template (e.g., thin film thickness); and(7) recalling the height value that provided the peak match positionwithin the data trace.

We now describe a suitable correlation technique based on a complexcorrelation. For each model of the test surface we generate a templateinterference patternI _(temp) ^(j)(ζ)=m _(temp) ^(j)(ζ)cos [K ₀ζ+φ_(temp) ^(j)(ζ)]  (108),where the index j indicates the specific model for the template pattern.The functions m_(temp) ^(j)(ζ) and φ_(temp) ^(j)(ζ) characterize thecomplex surface structure, but are independent of surface height at thelocation corresponding to the signal, which is set to zero. In preferredembodiments, the functions m_(temp) ^(j)(ζ) and φ_(temp) ^(j)(ζ) alsoaccount for systematic contributions from the interferometer. We thenuse a complex representation for the template pattern:{overscore (I)} _(temp) ^(j)(ζ)=m _(temp) ^(j)(ζ)exp [i(K ₀ζ+φ_(temp)^(j)(ζ))]  (109).We further use a window function to select a particular portion of thecomplex template function: $\begin{matrix}{{w(\zeta)} = \left\{ \begin{matrix}1 & {{{for}\quad\zeta_{start}} \leq \zeta \leq \zeta_{stop}} \\0 & {otherwise}\end{matrix} \right.} & (110)\end{matrix}$  {overscore (I)} _(pat) ^(k)(ζ)=w(ζ){overscore (I)}_(temp) ^(k)(ζ)  (111)For example, an appropriate window might be $\begin{matrix}{{\zeta_{start} = {- \frac{\Delta\quad\zeta}{2}}}{\zeta_{stop} = {+ \frac{\Delta\quad\zeta}{2}}}} & (112)\end{matrix}$where the window width Δζ could be set by hand.

Now that we have an interference pattern template Ĩ_(pat) ^(j) we areready to use it for comparison to an actual data set. In preparation forthis, it will be handy to generate a complex signal Ĩ_(ex) ^(j) startingfrom a real experimental data setI _(ex)(ζ,x)=DC _(ex)(x)+ . . . AC _(ex)(x)m _(ex) [ζ−h _(ex)(x)] cos{−[ζ−h _(ex)(x)]K ₀+φ_(ex) [ζ−h _(ex)(x)]}.  (113)The Fourier transform of this signal isq _(ex)(K,x)=FT{I _(ex)(ζ,x)}  (114)q _(ex)(K,x)=δ(K)DC _(ex)(x)+½AC _(ex)(x)[G* _(ex)(−K−K ₀ ,x)+G_(ex)(K−K ₀ ,x)]  (115)whereG _(ex)(K)=FT{m _(ex)(ζ)exp [iφ _(ex)(ζ)]} exp [iKh _(ex)(x)].  (116)We then construct a partial spectrum from the positive-frequency portionof the spectrum:{tilde over (q)} _(ex)(K)=AC _(ex)(x)G _(ex)(K−K ₀ ,x).  (117)The inverse transform is thenĨ _(ex)(ζ)=FT ⁻¹ {{tilde over (q)} _(ex)(K)}  (118)Ĩ _(ex)(ζ,x)=AC _(ex)(x)m _(ex) [ζ−h _(ex)(x)] exp {−i[ζ−h _(ex)(x)]K ₀+iφ _(ex) [ζ−h _(ex)(x)]}  (119)Here, the real part of this complex function Ĩ_(ex) is the originalexperimental data I_(ex). Further, the phase and envelope are separableby simple operations, e.g. we can access the product of the signalstrength AC_(ex)(x) and envelope m_(ex) using the magnitude of thecomplex function Ĩ_(ex):AC _(ex)(x)m _(ex) [ζ−h _(ex)(x)]=|Ĩ _(ex)(ζ,x)|.  (120)According to the underlying theory of the technique, we expect at leasta meaningful portion of m_(ex) to have the same general shape asm_(temp) ^(j) for the correct model, the only difference being thelinear offset h_(ex) and the scaling factor AC_(ex)(x). We also expectthe difference between the experimental and interference patterntemplate phase offsets φ_(ex), φ_(pat) ^(j), respectively, to belinearly proportional to the height h_(ex), for the correct model.

The task at hand is to locate a specific signal pattern represented bythe interference pattern template Ĩ_(pat) ^(j), within an experimentaldata set Ĩ_(ex), and determine how well of a match there is for each ofthe different models j. In what follows, we shall drop the index j, andnote the matching analysis proceeds for each of the models.

The first step is to find the scan position ζ_(best) for which theshapes of the envelopes m_(ex), m_(pat) and φ_(ex), φ_(pat) are bestmatched. A viable approach is a merit function based on the normalizedcorrelation of the interference pattern template with the signal withina segment of the scan defined by the window w: $\begin{matrix}{{\Pi\left( {\zeta,x} \right)} = {\frac{{{\overset{\sim}{I}\left( {\zeta,x} \right)}}^{2}}{\left\langle m_{pat}^{2} \right\rangle\left\langle {{{\overset{\sim}{I}}_{ex}\left( {\zeta,x} \right)}}^{2} \right\rangle}\quad{where}}} & (121) \\{{\overset{\sim}{I}\left( {\zeta,x} \right)} = {\frac{1}{N}{\int_{- \infty}^{\infty}{{{\overset{\sim}{I}}_{pat}^{*}\left( \hat{\zeta} \right)}{{\overset{\sim}{I}}_{ex}\left( {{\zeta + \hat{\zeta}},x} \right)}{{\mathbb{d}\hat{\zeta}}.}}}}} & (122)\end{matrix}$is the complex correlation function and $\begin{matrix}{\left\langle m_{pat}^{2} \right\rangle = {\frac{1}{N}{\int_{- \infty}^{\infty}{{{{\overset{\sim}{I}}_{pat}\left( \hat{\zeta} \right)}}^{2}{\mathbb{d}\hat{\zeta}}}}}} & (123) \\{\left\langle {{{\overset{\sim}{I}}_{ex}\left( {\zeta,x} \right)}}^{2} \right\rangle = {\frac{1}{N}{\int_{- \infty}^{\infty}{{{{\overset{\sim}{I}}_{ex}\left( {{\zeta + \hat{\zeta}},x} \right)}}^{2}{w\left( \hat{\zeta} \right)}{\mathbb{d}\hat{\zeta}}}}}} & (124)\end{matrix}$are normalizations that make the merit function Π independent ofsignal-strength. Use of the complex conjugate Ĩ*_(pat) of the templatecancels the synchronous linear phase term K₀ζ and maximizes Π for thecase of a match of φ_(ex), φ_(pat). The absolute value | | of thecorrelation removes any residual complex phase.

To prevent Π (ζ) from generating false high values or encountering asingularity at low signal levels, it is prudent to add a minimum valueto the denominator, such as

|Ĩ _(ex)(ζ)|² →|Ĩ _(ex)(ζ)|² +MinDenom·max(

|Ĩ _(ex)|²

)  (125)where the max( ) function returns the maximum value of the signalstrength |Ĩ_(ex)| over the full scan length ζ, and MinDenom is theminimum relative signal strength that we consider valid in the meritfunction search. The value of MinDenom can be hard coded at 5% or someother small value, or left as an adjustable parameter.

The correlation integral Ĩ can also be performed in the frequency domainusing the correlation theorem:Ĩ(ζ)=FT ⁻¹ {{tilde over (q)}* _(pat)(K){tilde over (q)} _(ex)(K)}  (126)where I have made use ofFT{Ĩ* _(pat)(ζ,x)}={tilde over (q)}* _(pat)(−K,x)  (127)where{tilde over (q)} _(pat)(K,x)=FT{Ĩ _(pat)(ζ,x)}.  (128)

A search through Π to find a peak value yields the best match positionζ_(best) and the value of Π is a measure of the quality of the match,ranging from zero to one, with one corresponding to a perfect match. Thepeak value of the merit function is calculated for each of the differentmodels to determine which model is the best match, and then the bestmatch position ζ_(best) for that model gives the surface height.

FIGS. 20-24 illustrate an example of the technique. FIG. 20 shows anactual scanning interferometry signal of a base Si substrate without athin film. FIGS. 21 and 22 show interference template patterns for abare Si substrate and a thin film structure with 1 micron of SiO2 on Si,respectively. FIGS. 23 and 24 show the merit function as a function ofscan positions for template functions in FIGS. 21 and 22, respectively.The merit functions show that the interference template pattern for thebare substrate is a much better match (peak value 0.92) than that forthe thin film template pattern (peak value 0.76) and therefore indicatethat the test sample is a bare substrate. Moreover, the position of thepeak in the merit function for the correct template pattern gives therelative surface height position for the test sample.

The methods and systems described above can be particularly useful insemiconductor applications. Additional embodiments of the inventioninclude applying any of the measurement techniques described above toaddress any of the semiconductor applications described below, andsystems for carrying out both the measurement techniques and thesemiconductor applications.

It is presently of considerable interest in the semiconductor industryto make quantitative measurements of surface topography. Due to thesmall size of typical chip features, the instruments used to make thesemeasurements typically must have high spatial resolution both paralleland perpendicular to the chip surface. Engineers and scientists usesurface topography measuring systems for process control and to detectdefects that occur in the course of manufacturing, especially as aresult of processes such as etching, polishing, cleaning and patterning.

For process control and defect detection to be particularly useful, asurface topography measuring system should have lateral resolutioncomparable to the lateral size of typical surface features, and verticalresolution comparable to the minimum allowed surface step height.Typically, this requires a lateral resolution of less than a micron, anda vertical resolution of less than 1 nanometer. It is also preferablefor such a system to make its measurements without contacting thesurface of the chip, or otherwise exerting a potentially damaging forceupon it, so as to avoid modifying the surface or introducing defects.Further, as it is well-known that the effects of many processes used inchip making depend strongly on local factors such as pattern density andedge proximity, it is also important for a surface topography measuringsystem to have high measuring throughput, and the ability to sampledensely over large areas in regions which may contain one or manysurface features of interest.

It is becoming common among chip makers to use the so-called ‘dualdamascene copper’ process to fabricate electrical interconnects betweendifferent parts of a chip. This is an example of a process which may beeffectively characterized using a suitable surface topography system.The dual damascene process may be considered to have five parts: (1) aninterlayer dielectric (ILD) deposition, in which a layer of dielectricmaterial (such as a polymer, or glass) is deposited onto the surface ofa wafer (containing a plurality of individual chips); (2) chemicalmechanical polishing (CMP), in which the dielectric layer is polished soas to create a smooth surface, suitable for precision opticallithography, (3) a combination of lithographic patterning and reactiveion etching steps, in which a complex network is created comprisingnarrow trenches running parallel to the wafer surface and small viasrunning from the bottom of the trenches to a lower (previously defined)electrically conducting layer, (4) a combination of metal depositionsteps which result in the trenches and vias being over-filled withcopper, and (5) a final chemical mechanical polishing (CMP) step inwhich the excess copper is removed, leaving a network of copper filledtrenches (and possibly vias) surrounded by dielectric material.

Typically the thickness of the copper in the trench areas (i.e., thetrench depth), and the thickness of the surrounding dielectric lie in arange of 0.2 to 0.5 microns. The width of the resulting trenches may bein a range of from 100 to 100,000 nanometers, and the copper regionswithin each chip may in some regions form regular patterns such asarrays of parallel lines, and in others they may have no apparentpattern. Likewise, within some regions the surface may be denselycovered with copper regions, and in other regions, the copper regionsmay be sparse. It is important to appreciate that the polishing rate,and therefore the remaining copper (and dielectric) thickness afterpolishing, depends strongly and in a complex manner on the polishingconditions (such as the pad pressure and polishing slurry composition),as well as on the local detailed arrangement (i.e., orientation,proximity and shape) of copper and surrounding dielectric regions.

This ‘position dependent polishing rate’ is known to give rise tovariable surface topography on many lateral length scales. For example,it may mean that chips located closer to the edge of a wafer onaggregate are polished more rapidly than those located close to thecenter, creating copper regions which are thinner than desired near theedges, and thicker than desired at the center. This is an example of a‘wafer scale’ process nonuniformity—i.e., one occurring on length scalecomparable to the wafer diameter. It is also known that regions whichhave a high density of copper trenches polish at a higher rate thannearby regions with low copper line densities. This leads to aphenomenon known as ‘CMP induced erosion’ in the high copper densityregions. This is an example of a ‘chip scale’ processnonuniformity—i.e., one occurring on a length scale comparable to (andsometimes much less than) the linear dimensions of a single chip.Another type of chip scale nonuniformity, known as ‘dishing’, occurswithin single copper filled trench regions (which tend to polish at ahigher rate than the surrounding dielectric material). For trenchesgreater than a few microns in width dishing may become severe with theresult that affected lines later exhibit excessive electricalresistance, leading to a chip failure.

CMP induced wafer and chip scale process nonuniformities are inherentlydifficult to predict, and they are subject to change over time asconditions within the CMP processing system evolve. To effectivelymonitor, and suitably adjust the process conditions for the purpose ofensuring that any nonuniformities remain within acceptable limits, it isimportant for process engineers to make frequent non-contact surfacetopography measurements on chips at a large number and wide variety oflocations. This is possible using embodiments of the interferometrytechniques described above.

Any of the computer analysis methods described above can be implementedin hardware or software, or a combination of both. The methods can beimplemented in computer programs using standard programming techniquesfollowing the method and figures described herein. Program code isapplied to input data to perform the functions described herein andgenerate output information. The output information is applied to one ormore output devices such as a display monitor. Each program may beimplemented in a high level procedural or object oriented programminglanguage to communicate with a computer system. However, the programscan be implemented in assembly or machine language, if desired. In anycase, the language can be a compiled or interpreted language. Moreover,the program can run on dedicated integrated circuits preprogrammed forthat purpose.

Each such computer program is preferably stored on a storage medium ordevice (e.g., ROM or magnetic diskette) readable by a general or specialpurpose programmable computer, for configuring and operating thecomputer when the storage media or device is read by the computer toperform the procedures described herein. The computer program can alsoreside in cache or main memory during program execution. The analysismethod can also be implemented as a computer-readable storage medium,configured with a computer program, where the storage medium soconfigured causes a computer to operate in a specific and predefinedmanner to perform the functions described herein.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention.

1. A method comprising: comparing information derivable from a scanninginterferometry signal for a first surface location of a test object toinformation corresponding to multiple models of the test object, whereinthe multiple models are parametrized by a series of characteristics forthe test object, wherein the derivable information being comparedrelates to a shape of the scanning interferometry signal for the firstsurface location of the test object.
 2. The method of claim 1, furthercomprising determining an accurate characteristic for the test objectbased on the comparison.
 3. The method of claim 1, further comprisingdetermining a relative surface height for the first surface locationbased on the comparison.
 4. The method of claim 3, wherein thedetermining of the relative surface height comprises determining whichmodel corresponds to an accurate one of the characteristic for the testobject based on the comparison, and using the model corresponding to theaccurate characteristic to calculate the relative surface height.
 5. Themethod of claim 4, wherein using the model corresponding to the accuratecharacteristic to calculate the relative surface height comprisesdetermining a position of a peak in a correlation function used tocompare the information for the test object to the information for themodel corresponding to the accurate characteristic.
 6. The method ofclaim 1, further comprising comparing information derivable from thescanning interferometry signal for additional surface locations to theinformation corresponding to the multiple models.
 7. The method of claim6, further comprising determining a surface height profile for the testobject based on the comparisons.
 8. The method of claim 1, wherein thecomparing comprises calculating one or more merit functions indicativeof a similarity between the information derivable from the scanninginterferometry signal and the information corresponding to each of themodels.
 9. The method of claim 1, wherein the comparing comprisesfitting the information derivable from the scanning interferometrysignal to an expression for the information corresponding to the models.10. The method of claim 1, wherein the information for the test objectrelates to a fringe contrast magnitude in the shape of the scanninginterferometry signal.
 11. The method of claim 1, wherein theinformation for the test object relates to relative spacings betweenzero-crossings in the shape of the scanning interferometry signal. 12.The method of claim 1, wherein the information for the test object is aexpressed as a function of scan position.
 13. The method of claim 12,wherein the comparing comprises calculating a correlation functionbetween the information for the test object and the information for eachof the models.
 14. The method of claim 12, wherein the comparing furthercomprises determining one or more peak values in each of the correlationfunctions.
 15. The method of claim 14, further comprising determining anaccurate characteristic for the test object based on theparameterization of the model corresponding to the largest peak value.16. The method of claim 14, further comprising determining a relativesurface height for the test object at the first surface location basedon a coordinate for at least one of the peak values in the correlationfunctions.
 17. The method of claim 1, wherein the scanninginterferometry signal is produced by a scanning interferometry system,and wherein the comparing comprises accounting for systematiccontributions to the scanning interferometry signal arising from thescanning interferometry system.
 18. The method of claim 17, furthercomprising calibrating the systematic contributions of the scanninginterferometry system using another test object having known properties.19. The method of claim 1, wherein the series of characteristicscomprises a series of values for at least one physical parameter of thetest object.
 20. The method of claim 19, wherein the test objectcomprises a thin film layer having a thickness, and the physicalparameter is the thickness of the thin film at the first location. 21.The method of claim 1, wherein the series of characteristics comprises aseries of characteristics of the test object at a second surfacelocation different from the first surface location.
 22. The method ofclaim 21, wherein the test object comprises structure at the secondsurface location that diffracts light to contribute to the scanninginterferometry signal for the first surface location.
 23. The method ofclaim 21, wherein the series of characteristics at the second surfacelocation comprises permutations of a magnitude for a step height at thesecond location and a position for the second location.
 24. The methodof claim 21, wherein the series of characteristics at the second surfacelocation comprises permutations of a modulation depth for a grating andan offset position of the grating, wherein the grating extends over thesecond location.
 25. The method of claim 1, wherein the series ofcharacteristics is a series of surface materials for the test object.26. The method of claim 1, wherein the models correspond to a fixedsurface height for the test object at the first surface location. 27.The method of claim 1, wherein the scanning interferometry signal isproduced by imaging test light emerging from the test object tointerfere with reference light on a detector, and varying an opticalpath length difference from a common source to the detector betweeninterfering portions of the test and reference light, wherein the testand reference light are derived from the common source, and wherein thescanning interferometry signal corresponds to an interference intensitymeasured by the detector as the optical path length difference isvaried.
 28. The method of claim 27, further comprising producing thescanning interferometry signal.
 29. The method of claim 27, wherein thetest and reference light have a spectral bandwidth greater than 5% of acentral frequency for the test and reference light.
 30. The method ofclaim 27, wherein the common source has a spectral coherence length, andthe optical path length difference is varied over a range larger thanthe spectral coherence length to produce the scanning interferometrysignal.
 31. The method of claim 27, wherein optics used to direct testlight onto the test object and image it to the detector define anumerical aperture for the test light greater than 0.8.
 32. The methodof claim 28, wherein the common source is a spatially extended source.33. Apparatus comprising: a computer readable medium having a programthat causes a processor in a computer to compare information derivablefrom a scanning interferometry signal for a first surface location of atest object to information corresponding to multiple models of the testobject, wherein the multiple models are parametrized by a series ofcharacteristics for the test object, wherein the derivable informationbeing compared relates to a shape of the scanning interferometry signalfor the first surface location of the test object.
 34. Apparatuscomprising: a scanning interferometry system configured to produce ascanning interferometry signal; and an electronic processor coupled tothe scanning interferometry system to receive the scanninginterferometry signal and programmed to compare information derivablefrom a scanning interferometry signal for a first surface location of atest object to information corresponding to multiple models of the testobject, wherein the multiple models are parametrized by a series ofcharacteristics for the test object, wherein the derivable informationbeing compared relates to a shape of the scanning interferometry signalfor the first surface location of the test object.
 35. A methodcomprising: comparing information derivable from a scanninginterferometry signal for a first surface location of a test object toinformation corresponding to multiple models of the test object, whereinthe multiple models are parametrized by a series of characteristics forthe test object, wherein the series of characteristics comprises aseries of characteristic of the test object at a second surface locationdifferent from the first surface location.
 36. The method of claim 35,wherein the test object comprises structure at the second surfacelocation that diffracts light to contribute to the scanninginterferometry signal for the first surface location.
 37. The method ofclaim 35, wherein the series of characteristics at the second surfacelocation comprises permutations of a magnitude for a step height at thesecond location and a position for the second location.
 38. The methodof claim 35, wherein the series of characteristics at the second surfacelocation comprises permutations of a modulation depth for a grating andan offset position of the grating, wherein the grating extends over thesecond location.
 39. The method of claim 35, further comprisingdetermining an accurate characteristic for the test object based on thecomparison.
 40. The method of claim 35, further comprising determining arelative surface height for the first surface location based on thecomparison.
 41. The method of claim 40, wherein the determining of therelative surface height comprises determining which model corresponds toan accurate one of the characteristic for the test object based on thecomparison, and using the model corresponding to the accuratecharacteristic to calculate the relative surface height.
 42. The methodof claim 41, wherein the using of the model corresponding to theaccurate characteristic comprises compensating data from the scanninginterferometry signal to reduce contributions arising from the accuratecharacteristic.
 43. The method of claim 42, wherein the compensating ofthe data comprises removing a phase contribution arising from theaccurate characteristic from a phase component of a transform of thescanning interferometry signal for the test object, and wherein theusing of the model corresponding to the accurate characteristic furthercomprises calculating the relative surface height from the phasecomponent of the transform after the phase contribution arising from theaccurate characteristic has been removed.
 44. The method of claim 42,wherein using the model corresponding to the accurate characteristic tocalculate the relative surface height comprises determining a positionof a peak in a correlation function used to compare the information forthe test object to the information for the model corresponding to theaccurate characteristic.
 45. The method of claim 35, further comprisingcomparing information derivable from the scanning interferometry signalfor additional surface locations to the information corresponding to themultiple models.
 46. The method of claim 45, further comprisingdetermining a surface height profile for the test object based on thecomparisons.
 47. The method of claim 35, wherein the comparing comprisescalculating one or more merit functions indicative of a similaritybetween the information derivable from the scanning interferometrysignal and the information corresponding to each of the models.
 48. Themethod of claim 35, wherein the comparing comprises fitting theinformation derivable from the scanning interferometry signal to anexpression for the information corresponding to the models.
 49. Themethod of claim 35, wherein the information derivable from the scanninginterferometry signal and which is being compared is a number.
 50. Themethod of claim 35, wherein the information derivable from the scanninginterferometry signal and which is being compared is a function.
 51. Themethod of claim 50, wherein the function is a function of spatialfrequency.
 52. The method of claim 50, wherein the function is afunction of scan position.
 53. The method of claim 35, wherein theinformation for the test object is derived from a transform of thescanning interferometry signal for the test object into a spatialfrequency domain.
 54. The method of claim 53, wherein the transform is aFourier transform.
 55. The method of claim 53, wherein the informationfor the test object comprises information about an amplitude profile ofthe transform.
 56. The method of claim 53, wherein the information forthe test object comprises information about a phase profile of thetransform.
 57. The method of claim 35, wherein information for the testobject relates to a shape of the scanning interferometry signal for thetest object at the first location.
 58. The method of claim 57, whereinthe information for the test object relates to a fringe contrastmagnitude in the shape of the scanning interferometry signal.
 59. Themethod of claim 57, wherein the information for the test object relatesto relative spacings between zero-crossings in the shape of the scanninginterferometry signal.
 60. The method of claim 57, wherein theinformation for the test object is a expressed as a function of scanposition, wherein the function is derived from the shape of the scanninginterferometry signal.
 61. The method of claim 35, wherein the comparingcomprises calculating a correlation function between the information forthe test object and the information for each of the models.
 62. Themethod of claim 61, wherein the correlation function is a complexcorrelation function.
 63. The method of claim 61, wherein the comparingfurther comprises determining one or more peak values in each of thecorrelation functions.
 64. The method of claim 63, further comprisingdetermining an accurate characteristic for the test object based on theparameterization of the model corresponding to the largest peak value.65. The method of claim 63, further comprising determining a relativesurface height for the test object at the first surface location basedon a coordinate for at least one of the peak values in the correlationfunctions.
 66. The method of claim 35, wherein the scanninginterferometry signal is produced by a scanning interferometry system,and wherein the comparing comprises accounting for systematiccontributions to the scanning interferometry signal arising from thescanning interferometry system.
 67. The method of claim 66, furthercomprising calibrating the systematic contributions of the scanninginterferometry system using another test object having known properties.68. Apparatus comprising: a computer readable medium having a programthat causes a processor in a computer to compare information derivablefrom a scanning interferometry signal for a first surface location of atest object to information corresponding to multiple models of the testobject, wherein the multiple models are parametrized by a series ofcharacteristics for the test object, wherein the series ofcharacteristics comprises a series of characteristic of the test objectat a second surface location different from the first surface location.69. Apparatus comprising: a scanning interferometry system configured toproduce a scanning interferometry signal; and an electronic processorcoupled to the scanning interferometry system to receive the scanninginterferometry signal and programmed to compare information derivablefrom a scanning interferometry signal for a first surface location of atest object to information corresponding to multiple models of the testobject, wherein the multiple models are parametrized by a series ofcharacteristics for the test object, wherein the series ofcharacteristics comprises a series of characteristic of the test objectat a second surface location different from the first surface location.70. A method comprising: chemically mechanically polishing a testobject; collecting scanning interferometry data for a surface topographyof the test object; and adjusting process conditions for the chemicallymechanically polishing of the test object based on information derivedfrom the scanning interferometry data.
 71. The method of claim 70,wherein adjusting the process conditions based on the informationderived from the scanning interferometry data comprises comparinginformation derivable from the scanning interferometry signal for atleast a first surface location of a test object to informationcorresponding to multiple models of the test object, wherein themultiple models are parametrized by a series of characteristics for thetest object.
 72. The method of claim 70, where the process conditionscomprise at least one of pad pressure and polishing slurry composition.